Download Topology I with a categorical perspective

Tai-Danae Bradley and John Terilla
Topology I
with a categorical perspective
October 10, 2016
Contents
0
Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi
0.1 Topological Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi
0.2 Basic category theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii
0.2.1 Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii
0.2.2 Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv
0.2.3 Natural transformations and the Yoneda lemma . . . . . . . . . . . xvi
0.3 Basic set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii
0.3.1 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii
0.3.2 The emptyset and one point set . . . . . . . . . . . . . . . . . . . . . . . . . xviii
0.3.3 Products and coproducts in Set . . . . . . . . . . . . . . . . . . . . . . . . . xviii
0.3.4 Products and coproducts in any category . . . . . . . . . . . . . . . . . xx
0.3.5 Exponentiation in Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxi
0.3.6 Partially ordered sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxi
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxi
1
Examples and constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1 Examples and terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1.1 Examples of spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1.2 Examples of continuous functions . . . . . . . . . . . . . . . . . . . . . .
1.2 The subspace topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.1 First characterization of the subspace topology . . . . . . . . . . .
1.2.2 Second characterization of the subspace topology . . . . . . . . .
1.3 The quotient topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3.1 The first characterization of the quotient topology . . . . . . . . .
1.3.2 The second characterization of the quotient topology . . . . . .
1.4 The product topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4.1 First characterization of the product topology . . . . . . . . . . . . .
1.4.2 Second characterization of the product topology . . . . . . . . . .
1.5 The coproduct topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.5.1 The first characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.5.2 The second characterization . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1
1
3
4
4
5
7
8
8
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12
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Contents
1.6
1.7
Homotopy and the homotopy category . . . . . . . . . . . . . . . . . . . . . . . . . 13
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2
Connectedness and compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1 Connectedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.1 Definitions, theorems, and examples . . . . . . . . . . . . . . . . . . . .
2.1.2 The functor π0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.3 Constructions and connectedness . . . . . . . . . . . . . . . . . . . . . . .
2.1.4 Local (path) connectedness . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Hausdorff spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.1 Definitions, theorems and examples . . . . . . . . . . . . . . . . . . . . .
2.3.2 Constructions and compactness . . . . . . . . . . . . . . . . . . . . . . . .
2.3.3 Local compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
17
17
20
21
22
23
24
24
25
26
3
Limits of sequences and nets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1 Closure and interior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 Nets and three theorems about them . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4 Tychonoff’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.1 Preliminaries from set theory . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.2 Nets and compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.3 A proof of Tychonoff’s Theorem . . . . . . . . . . . . . . . . . . . . . . .
3.4.4 Tychonoff’s theorem implies the axiom of choice . . . . . . . . .
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
31
31
35
36
37
38
40
40
41
4
Categorical limits and colimits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1 Diagrams are functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Limits and colimits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.1 Initial and terminal objects . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.2 Pushouts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.3 Pullbacks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.4 Equalizers and coequalizers . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.5 Direct and inverse limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4 Completeness and cocompleteness . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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50
5
Adjunctions and the compact open topology . . . . . . . . . . . . . . . . . . . . . . .
5.1 Adjunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1.1 The unit and counit of an adjunction . . . . . . . . . . . . . . . . . . . .
5.2 Free-Forgetful adjunction in algebra . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3 Compactifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3.1 The one-point compactification . . . . . . . . . . . . . . . . . . . . . . . .
5.3.2 The Stone-Čech compactification . . . . . . . . . . . . . . . . . . . . . . .
5.4 The forgetful functor U : Top → Set and its adjoints . . . . . . . . . . . . .
5.5 The exponential topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Contents
5.5.1 The compact-open topology . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.5.2 The theorems of Ascoli and Arzela . . . . . . . . . . . . . . . . . . . . .
5.6 The compact-open topology when X is locally compact Hausdorff .
5.6.1 Lemmas about normal spaces . . . . . . . . . . . . . . . . . . . . . . . . . .
5.6.2 The compact open topology is exponential . . . . . . . . . . . . . . .
5.6.3 Enrich the product-hom adjunction in Top . . . . . . . . . . . . . . .
5.7 Compactly generated weakly Hausdorff spaces . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Preface
When teaching a graduate topology course, it’s tempting to rush through the pointset topology, or even skip it altogether, and do more algebraic topology, which is
more fun to teach and more relevant to today’s students. One gets away with it
because many point set topology ideas are already familiar to students from undergraduate analysis or elementary point-set topology courses, and seems safe to skip.
Also, point-set ideas that might be unfamiliar but important in other subjects, say
the Zariski topology in algebraic geometry or the p-adic topology in number theory,
aren’t too difficult to pick up later whenever and wherever they are encountered.
An alternative to rushing through point-set topology is to cover it from a more
modern, categorical point of view. There are a number of reasons this alternative
can be better. Since many students are familiar with point-set ideas already, they
are in a good position to learn something new about these ideas, like the universal
properties characterizing them. Plus, using categorical methods to handle point-set
topology, whose name even suggests an old-fashioned way of thinking of spaces,
demonstrates the power and versatility of the methods. The category of topological
spaces is poorly behaved in some respects, but this provides opportunities to draw
meaningful contrasts between topology and other subjects, and to give good reasons
why certain kinds of spaces (like compactly generated spaces or CW complexes)
enjoy such prevalence. Finally, there is the practicality that point set topology is
on the syllabus for our first year topology course and PhD exam. So teaching the
material in a way that both deepens understanding and prepares a solid foundation
for future work in modern mathematics, is an excellent alternative.
This text contains material currated from many resources in order to present elementary topology from a categorical perspective. The result is intentionally less
comprehensive but hopefully more useful. It’s assumed that students know linear
algebra well and have had at least enough abstract algebra to understand how to
form the quotient of a group by a normal subgroup. Students should also have some
basic knowledge about how to work with sets and their elements, even as they endeaver to work with arrows instead. Students encountering diagrams and arrows for
the first time may want to spend a little extra time reading the preliminaries where
the objects (sets) are presumably familiar but the perspective may be new.
ix
Chapter 0
Preliminaries
I argue that set theory should not be based on membership, as in Zermelo-Frankel set theory,
but rather on isomorphism-invariant structure.
— William Lawvere
We’ve assembled some preliminary material is here. We assume that the reader
is familiar with some, but probably not all of this material.
0.1 Topological Spaces
Definition 0.1. A topological space consists of a set X and a collection τ of subsets
of X, called open sets, satisfying the following properties:
• The sets ∅ and X are in τ.
• Any union of elements in τ is also in τ.
• Any finite intersection of elements in τ is also in τ.
The collection τ is called a topology on X. A set C is called closed if its complement
is open.
Example 0.1. Let X be any set. The collection 2 X of all subsets of X forms a topology called the discrete topology on X. The set {∅, X } forms a topology on X called
the indiscrete topology or the trivial topology.
Sometimes, two topologies on the same space are comparable. When τ ⊆ τ 0, the
topology τ can be called coarser than τ 0, or the topology τ 0 can be called finer than
τ. Instead of courser and finer, some people say “smaller” and “larger” or “stronger”
and “weaker” but the terminology becomes clearer—as with most things in life—
with coffee. A coarse grind yields a small number of chunky coffee pieces, whereas
a fine grind results in a large number of tiny coffee pieces. Finely ground beans
make stronger coffee, coarsely ground beans make weaker coffee.
xi
xii
0 Preliminaries
In practice, it can be easier to work with a small collection of open subsets of X
that generates the topology.
Definition 0.2. A collection B of subsets of a set X is a basis for a topology on X
if and only if
• For each x ∈ X there is a B ∈ B such that x ∈ B.
• If x ∈ B1 ∩ B2 where B1 , B2 ∈ B, then there is at least one B3 ∈ B such that
x ∈ B3 ⊆ B1 ∩ B2 .
The topology τ generated by the basis B is defined to be the coarsest topology
containing B. Equivalently, a set U ⊆ X is open in the topology generated the basis
B if and only if for every x ∈ U there is a B ∈ B such that x ∈ B ⊆ U.
Example 0.2. A metric space is a pair (X, d) where X is a set and d : X × X → R
satisfies
•
•
•
•
d(x, y) ≥ 0 for all x, y ∈ X,
d(x, y) = d(y, x) for all x, y ∈ X,
d(x, y) + d(y, z) ≤ d(x, z) for all x, y, z ∈ X
d(x, y) = 0 if and only if x = y for all x, y ∈ X.
The function d is called a metric or a distance function. If (X, d) is a metric space,
x ∈ X, and r > 0, the ball centered at x of radius r is defined to be
B(x,r) = {y ∈ X : d(x, y) < r } .
The balls {B(x,r)} form a basis for a topology on X called the metric topology.
Any subset of a metric space is a metric space. Since Rn with the usual Euclidean
distance function is a metric space, subsets of Rn provide numerous examples of
topological spaces. In particular the real line R, the unit interval I := [0, 1], the
closed unit ball D n := {(x 1 , . . . , x n ) ∈ Rn : x 21 + · · · + x 2n ≤ 1}, and the n-sphere
S n := {(x 1 , . . . , x n+1 ) ∈ Rn+1 : x 21 + · · · + x 2n+1 = 1}are important topological
spaces.
Definition 0.3. A function f : X → Y between two topological spaces is continuous
if and only if for every open set U ⊆ Y , f −1 (U) is open in X.
It is straightforward to check that for any topological space X, the idenity id X :
X → X is continuous and that for any topological spaces X,Y, Z and any continuous
functions f : X → Y and g : Y → Z, the composition g ◦ f : X → Z is continuous.
Thus, topological spaces and continuous functions form a category.
0.2 Basic category theory
0.2.1 Categories
Definition 0.4. A category C consists of the following data:
0.2 Basic category theory
xiii
• A class of objects,
• For every two objects X,Y , there is a set1 hom(X,Y ) called morphisms. The
expression f : X → Y means f ∈ hom(X,Y ).
• There is a composition rule defined for morphisms. For f : X → Y and g : Y →
Z there is a morphism g f : X → Z.
These data must satisfy the following two conditions:
• Composition is associative. That is, if h : X → Y , g : Y → Z, f : Z → W then
f (gh) = ( f g)h. Here’s the picture:
gh
←
←
→Y
g
→ Z
←
←
←
X
→
h
f
→ W
→
fg
• There exist identity morphisms. That is, for every object X, there exists a morphism id X : X → X satisfying the property that f id X = f = idY f whenever
f : X → Y.
By the usual argument identity morphisms are unique. Some examples of categories
are
• Set: the objects are sets, the morphisms are functions, composition is composition of functions.
• Vectk : the objects are vector spaces over a fixed field k, the morphisms are linear
transformations, composition is composition of linear transformations.
• RMod: Fix a ring R. The objects are R-modules, the morphisms are R-module
maps, and composition is composition of module maps.
• Group: the objects are groups, the morphisms are group homomorphisms, and
composition is composition of homomorphisms.
• Let G be a group. Define a category C with one object ∗ and with hom(∗, ∗) = G
with composition defined as in the group G.
• Set∗ : the objects are pointed sets (sets together with a distinguished element), the
morphisms are functions that respect the distinguished elements, composition is
composition of functions.
• Top: the objects are topological spaces, the morphisms are continuous functions,
and composition is composition of functions.
• Top∗ : the objects are pointed topological spaces, the morphisms are continuous
functions that respect the basepoints, and composition is composition of functions.
• hTop. The objects are topological spaces, the morphisms are homotopy classes
of continuous functions. Homotopy is defined later.
• A directed multigraph defines a category whose objects are the nodes. The morphisms are the directed paths.
1 Having a set’s worth of morphisms means our categories are locally small.
xiv
0 Preliminaries
• For any category C, there is an opposite category Cop whose objects are the same
as the objects of C but whose morphisms are reversed. Composition in Cop is
defined by composition in C. That is homCop (X,Y ) = homC (Y, X ) and if f ∈
homCop (X,Y ) and g ∈ homCop (Y, Z ), the composition is f g ∈ homCop (X, Z )
which makes sense since g : Z → Y and f : Y → X implies f g : Z → X as is
necessary.
The fact that the examples above form categories should be verified. For example, in Vect, it should be checked that the composition of linear transformations is
again a linear transformation. Associativity of composition is automatic since linear
transformations are functions and composition of functions is always associative.
And for any vector space, the identity function is a linear transformation.
Definition 0.5. Let X,Y be objects in any category. A morphism f : X → Y is
called an isomorphism if there exists a morphism g : Y → X with g f = id X and
f g = idY . Two objects X and Y are isomorphic, denoted X Y , if there exists an
isomorphism f : X → Y .
Isomorphic is always reflexive, symmetric, and transitive and isomorphic objects
form equivalence classes. Sometimes categories have their own special terminology. The isomorphisms in Top are also called homeomorphisms. Two sets X and Y
that are isomorphic are said to have the same cardinality and a cardinal is an isomorphism class of sets. Mathematics in a category C is concerned with isomorphism
invariant concepts. For example, a property P is a topological property if and only
if whenever a space X has (or doesn’t have) property P and Y is isomorphic to X,
then Y has (or doesn’t have) property P.
Example 0.3. The cardinality of a topological space is a topological property since if
f : X → Y is a homeomorphism, it is an invertible function, so as sets X and Y have
the same cardinality. Connected, compact, Hausdorff, metrizable, first countable, ...
are examples of other topological properties that appear later.
Example 0.4. A metric space is called complete if every Cauchy sequence converges. Being a complete metric space is not a topological property. The map
x 7→ x/(1 − x 2 ) is a homeomorphism (−1, 1) −
→ R, but R is a complete metric
space and (−1, 1) is not a complete metric space. A metric space is called bounded
if the metric is a bounded function. Being bounded is not a topological property, as
the previous example proves.
For each morphsim f : X → Y in a category, there is a map of sets f ∗ :
hom(Z, X ) → hom(Z,Y ) called the pushforward of f defined by postcomposition
f ∗ : g 7→ f g.
f
←
hom(Z, X )
←
X
→Y
f∗
→ hom(Z,Y )
0.2 Basic category theory
xv
There is also a map of sets f ∗ : hom(Y, Z ) → hom(X, Z ) called the pullback defined
by precomposition f ∗ : g 7→ g f .
f
←
hom(X, Z )
→Y
f∗
→
X
← hom(Y, Z )
The following theorem clarifies the maxim of category theory that objects are determined by their relationships with other objects.
Theorem 0.1. A morphism f : X → Y is an isomorphism if and only if for every
object Z, the pushforward f ∗ : hom(Z, X ) → hom(Z,Y ) is an isomorphism of sets
if and only if for every object Z, the pullback f ∗ : hom(Y, Z ) → hom(X, Z ) is an
isomorphism of sets.
Proof. We’ll prove that a morphism f : X → Y is an isomorphism if and only if for
every object Z, the pushforward f ∗ : hom(Z, X ) → hom(Z,Y ) is an isomorphism
of sets and leave the other statement as an exercise. Suppose f : X → Y is an
isomorphism. Let g : Y → X be the inverse of f . Note that for any Z, the map
g∗ : hom(Z,Y ) → hom(Z, X ) is the inverse of f ∗ .
Conversely, suppose that for any Z, the map f ∗ : hom(Z, X ) → hom(Z,Y ) is
an isomorphism of sets. For Z = Y , we have f ∗ : hom(Y, X ) −
→ hom(Y,Y ), in
particular f ∗ is surjective. Therefore, there exists a morphism g : Y → X so that
f ∗ g = idY . This means that f g = idY . To see that g f = id X , look at when Z = X.
We know f ∗ : hom(X, X ) −
→ hom(X,Y ), in particular f ∗ is injective. Note that
f ∗ (id X ) = f and f ∗ (g f ) = f g f = f , therefore id X = g f as needed.
0.2.2 Functors
Definition 0.6. A functor F from a category C to a category D consists of the following data:
• An object F X of the category D for each object X in the cateogry C,
• A morphism F f : F X → FY for every morphism f : X → Y .
These data must must be compatible with composition and identity morphisms:
• (Fg)(F f ) = F (g f ) for any morphisms f : X → Y and g : Y → Z,
• F (id X ) = id F X for any object X.
Here are some examples of functors.
• For an object X in a category C, there is a functor h X := hom(X, −) from C to
Set defined on objects by h X (Z ) = hom(X, Z ) and on morphisms by h X f = f ∗ .
xvi
0 Preliminaries
• For an object X in a category C, there is a functor h X := hom(−, X ) from Co p to
Set defined on objects by h X (Z ) = hom(Z, X ) and on morphisms by h X f = f ∗ .
• The fundamental group defines a functor from Top∗ to Group. This will be discussed in detail in Chapter ??.
• There is a forgetful functor, usually denoted U for “underlying,” from Group to
Set that forgets the group operation.
• There is a free functor F from Set to Group that assigns the free group F S to the
set S.
• Fix a set Y . There is a functor ×Y from Set to Set defined on objects by X 7→
X × Y and on morphisms f by f × id .
• Fix a vector space V over a field k. There is a functor ⊗V from Vectk to Vectk
defined on objects by W 7→ W ⊗ V and on morphisms f by id ⊗ f .
• The construction of the Grothendieck group of a commutative monoid is functorial. That is, there is a “Grothendieck group” functor from the category of commutative monoids to the category of commutative groups that constructs a group
from a commutative monoid by attaching inverses.
One of the fundamental ideas in category theory is that because functors respect
composition and identities, functors take isomorphisms to isomorphisms and therefore define isomorphism invariants of objects. The main idea of algebraic topology
is to define functors from the category Top to algebraic categories. For example, a
homology theory is a functor Top → RMod and is therefore a means of distinguishing topological spaces. If H (X ) 6 H (Y ) then X and Y are not isomorphic.
Definition 0.7. Let F be a functor from a category C to a category D. For any objects
X and Y in C, there is a map
homC (X,Y ) → homD (F X, FY ).
The functor F is called faithful if this map is injective, it is called full if this map is
surjective, and it is called fully faithful if this map is a bijection.
0.2.3 Natural transformations and the Yoneda lemma
Definition 0.8. Let F and G are functors between the categories C and D. A natural
transformation η from F to G assigns a morphism η X : F (X ) → G(X ) to each
object X in C. These morphisms in D must satisfy the following property: for every
morphism f : X → Y in C, ηY F ( f ) = G( f )η X . Here’s the picture
F(f )
←
→ FY
←
←
FX
←
GX
→
ηY
→
ηX
→ GY
G( f )
0.3 Basic set theory
xvii
For any two functors F, G : C → D, let N at(F, G) denote the natural transformations
from F to G. If η X : F X −
→ GX is an isomrphism for each X, then η is called a
natural isomorphism, or a natural equivalence.
op
For any category C, there is a category SetC whose objects are functors from
op
C to Set and whose morphisms are natural transformations. Functors Cop → Set
op
are sometimes called pre-sheaves on C. The category SetC of presheaves on C is
a very nice category—it has all finite limits and colimits, it is Cartesian closed, it
forms what’s called a topos. We won’t dwell upon these properties further here, but
here’s one result that is good to know about.
op
The Yoneda Lemma. For every object X in C and for every functor F ∈ SetC , the
set of natural transformations from F to h X is isomorphic to F (X ).
The Yoneda lemma has some interesting corollaries. Setting the functor F = hY in
the Yoneda lemma shows that N at(hY , h X ) hY (X ) = hom(X,Y ) and illustrates
one significance of the Yoneda lemma. The assignment X 7→ h X defines a fully
op
faithful functor from C to SetC called the Yoneda embedding. So the Yoneda lemma
shows that every category C can be fully and faithfully viewed as a subcategory of
op
a very nice category, namely the category of presheaves SetC .
0.3 Basic set theory
We assume the reader has a basic working knowledge of sets and functions. Some
of the basics are reviewed here.
0.3.1 Functions
A function is injective if and only if it is left-cancellative. That is, f : X → Y
is injective if and only if for all functions g1 , g2 : Z → X with f g1 = f g2 it
follows that g1 = g2 . That is, f is injective if and only if f ∗ : hom(Z, X ) →
hom(Z,Y ) is injective for all Z. Left-cancellative morphisms in any category are
called monomorphisms or said to be monic and are denoted with arrows with tails
as in X Y . In Set, injective functions will be denoted by hooked arrows like this
f : X ,→ Y . In fact, a function is injective if and only if it has a left inverse. That
is, f : X → Y is injective if and only if there exists g : Y → X so that g f = idY .
The composition of injective functions is injective. Also, for any f : X → Y and
g : Y → Z, if g f is injective then f is injective.
Left-invertible implies left-cancellative in any category, but not conversely. For
example, the map n 7→ 2n defines an left-cancellative group homomorphism f :
Z/2Z → Z/4Z. However, there is no group homomorphism g : Z/4Z → Z/2Z so
that g f = idZ/2Z .
xviii
0 Preliminaries
A function is surjective if and only if it is right-cancellative. That is, f : X → Y
is surjective if and only if for all functions g1 , g2 : Y → Z with g1 f = g2 f it
follows that g1 = g2 . That is, f is surjective if and only if f ∗ : hom(Y, Z ) →
hom(X, Z ) is injective for all Z. Right-cancellative morphisms in any category are
called epimorphisms or said to be epic and are denoted with two-headed arrows as
in X Y . In Set, surjective functions will be denoted this way with two headed
arrows. In fact, a set function is surjective if and only if it has a right inverse. That
is, f : X → Y is surjective if and only if there exists g : Y → X so that f g = id X .
The composition of surjective functions is surjective. Also, for any f : X → Y and
g : Y → Z, if g f is surjective then g is surjective.
Right-invertible implies right-cancellative in any category, but not conversely. In
Set, a function that is both injective and surjective is an isomorphism. This is because left-invertible and right-invertible imply invertible (check that having a left
inverse and a right inverse both imply there’s a single two sided inverse). Leftcancellative and right-cancellative together do not imply invertible: there are categories (and Top is one of them) which have morphisms that are both monic and
epic and fail to be isomorphisms.
0.3.2 The emptyset and one point set
The empty set ∅ is initial in Set. That is, for any set X, there is a unique function
∅ → X. The one point set ∗ is terminal. That is, for any set X, there is a unique
function X → ∗. The reader may take issue with the definite article “the” in “the
one point set,” but it is standard to use the definite article in circumstances that are
unique up to unique isomorphism. That is the case here: if ∗ and ∗0 are both one
∼
point sets then there is a unique isomorphism ∗ −
→ ∗0. The notions of initial and
terminal objects make sense for any category, though such objects may not exist.
0.3.3 Products and coproducts in Set
The Cartesian product of two sets X and Y is a set X × Y that comes with with maps
π1 : X × Y → X and π2 : X × Y → Y . The product is characterized by the property
that for any set Z and any functions f 1 : Z → X and f 2 : Z → Y , there is a unique
map h : Z → X × Y with π1 h = f 1 and π2 h = f 2 . Here’s a picture
0.3 Basic set theory
xix
←
←
←
Z
h
f2
→
f1
←
π1
π2
→
→
→
X ×Y
←
→
X
Y
As an example, note {1, . . . , n} × {1, . . . , m} {1, . . . , nm}.
`
The disjoint union of two sets X and Y is a set X Y that comes with maps
`
`
i 1 : X → X Y and i 2 : Y → X Y . The disjoint union is characterized by the
property that for any set Z and any functions f 1 : X → Z and f 2 : Y → Z, there is
`
a unique map h : X Y → Z with hi 1 = f 1 and hi 2 = f 2 . Here’s a picture
Y
←
i1
`
→
X
→
i2
←
←
X
Y
h
←
→
→
←
→
f1
f2
Z
Xα
←
←
iα
←
→
Sometimes, disjoint union is called the sum and denoted X + Y or X ⊕ Y instead of
`
X Y . As an example, note {1, . . . , n} + {1, . . . , m} {1, . . . , n + m}. The property
characterizing the disjoint union is dual to the one characterizing the product and
disjoint union is sometimes called the coproduct of sets.
One can take products and disjoint unions of arbitrary collections of sets. The
`
disjoint union of a collection of sets {Xα }α ∈ A is a set α ∈ A Xα together with maps
`
i α : Xα → α ∈ A Xα satisfying the property that for any set Z and any collection of
`
functions { f α : Xα → Z }, there is a unique map h : α ∈I Xα → Z with hi α = f α
for all α ∈ A.
`
α ∈ A Xα
h
fα
→
→ Z
The product of a collection of sets {Xα }α ∈ A is sometimes described as the subset
`
of functions f : A → Xα satisfying f (α) ∈ Xα . What’s more important than the
construction of the product is to understand its universal property. The product of a
Q
Q
collection of sets {Xα }α ∈ A is a set α ∈ A Xα together with maps πα : α ∈ A Xα →
Xα characterized by the property that for any set Z and any collection of functions
Q
{ f α : Z → Xα }, there is a unique map h : Z → α ∈ A Xα with πα h = f α for all
α ∈ A.
0 Preliminaries
←
←
Z
fα
α∈A
→
Xα
πα
→
h
Q
←
xx
→ Xα
0.3.4 Products and coproducts in any category
Products and coproducts can be defined in any category using the universal properties described above. A more complete discussion of products and coproducts and
more general limits and colimits is in Chapter 4. For now, we’d like to point out
that in an arbitrary category, products and coproducts may not exist, and when they
do they might not look like disjoint unions or Cartesian products. For example, the
category Field of fields doesn’t have products: if there were a field F that were the
product of F2 and F3 , there would be homomorphisms F → F2 and F → F3 , which
is impossible since the characteristic of F would be a divisor of both 2 and 3. The
category Vect and more generally RMod has both products and coproducts. Products are Cartesian products but coproducts are direct sums. In Group coproducts are
free products but in the category of abelian groups, coproducts are direct sums. Even
in the category Set, there is something to say about the existence of products and
coproducts. The axiom of choice is precisely the statement that for any nonempty
Q
collection of sets {Xα }α ∈ A , the product α ∈ A Xα exists and is nonempty. Coproducts in Set are also axiomatically guaranteed to exist, by the axioms of union and
extension.
Even though products and coproducts in an arbitrary category might look different than they do in Set, these constructions are related to products and coproducts
in Set because the universal properties of products and coproducts yield bijections
of sets
hom *
,
a
α
Xα , Z + -
Y
α
hom (Xα , Z ) and hom * Z,
,
Y
α
Xα + -
Y
hom (Z, Xα ) .
α
Try to remember that coproducts come out of the first entry of hom as products,
and products come out of the second entry of hom as products. An example where
this comes up a lot is in Vect and RMod where the coproduct is direct sum and the
product is Cartesian product. For an R module X, let X ∗ := hom(X, R) denote the
dual space. Then setting Z = R in the first isomorphism above yields
M ∗ Y
Xα (Xα ) ∗
Remember, the dual of the sum is the product of the duals.
0.3 Basic set theory
xxi
0.3.5 Exponentiation in Set
In the category of sets, hom(X,Y ) is also denoted Y X . There is a natural evaluation
map ev : Y X × X → Y defined by ev( f , x) = f (x). The exponential notation is
convenient for expressing various isomorphisms, such as
(X × Y ) Z X Z × Y Z
which is an expression of the property characterizing products: maps from a set
Z into a product correspond to maps from Z into the factors. There is also the
isomorphism
Y X ×Z (Y X ) Z
which is an expression of the ×-hom adjunction. Fix a set X. Let F be the functor
X ×− and let G be the functor hom(X, −). In this notation, Y X ×Z (Y X ) Z becomes
hom(F Z,Y ) hom(Z, GY ) and evokes of the defining property of adjoint linear
maps.
0.3.6 Partially ordered sets
A partially ordered set or poset is a set P together with a relation ≤ on P that is
reflexive, transitive, and antisymmetric. Reflexive means that for all a ∈ P, a ≤ a;
transitive means that for all a, b, c ∈ P, if a ≤ b and b ≤ c then a ≤ c; antisymmetric
means that for all a, b ∈ P, if a ≤ b and b ≤ a then a = b.
One can view a poset as a category whose objects are the elements of P and
with a morphism a → b if and only if a ≤ b. Transitivity says composition can be
defined, and defined in only one way since there’s at most one morphism between
objects. Alternatively, one can define a poset to be a category with the property that
there’s at most one morphism between objects.
Exercises
0.1. Suppose S is a collection of subsets of X whose union equals X. Prove there is
a coursest topology τ containing S and that the collection of all finite intersections
of sets in S is a basis for τ. In this situation, the collection S is called a sub-basis
for the topology τ.
0.2. Prove that a function f : X → Y between topological spaces is continuous if
and only if f −1 (B) is open for every B in a basis for the topology on Y .
0.3. Examples and short proofs about morphisms.
xxii
0 Preliminaries
• Prove that left-invertible morphisms are monic and right-invertible morphisms
are epic.
• Give an example of a morphism that is epic but not right invertible.
• Prove that if a morphism is left-invertible and right-invertible then it is invertible.
• Give an example of a morphism that is epic and monic but not an isomorphism.
• Give an example of two objects X and Y that are not isomorphic and monomorphisms X Y and Y X.
0.4. Discuss the initial object, the terminal object, products, and coproducts in the
categories Group and Vect.
0.5. Prove the other part of Theorem 0.1. That is, prove that f : X → Y is an
isomorphism if and only if f ∗ : hom(Z, X ) → hom(Z,Y ) is an isomorphism for
every object Z.
0.6. Prove the Yoneda lemma. The key is to observe that h X (X ) has a special element, namely id X . So, for any natural tranformaiton η : h X → F, one obtains a
special element η(id X ) ∈ F (X ) which completely determines η.
Chapter 1
Examples and constructions
All of it was written by Sammy! I wrote nothing.
— Henri Cartan
This chapter contains four ways to construct new topological spaces from old ones:
subspaces, quotients, products, and coproducts. Naturally, these constructions are
more relevant when one is familiar with a few topological spaces and continuous
functions to use in the constructions.
1.1 Examples and terminology
We give some examples of spaces, then some examples of continuous functions.
1.1.1 Examples of spaces
Example 1.1. Any set X has a cofinite topology where a set U is open if and only
X \ U is finite (or if U = ∅). The open sets in the cocountable topology are those
whose complement is countable.
Example 1.2. The empty set ∅ and the one-point set ∗ are topological spaces in
unique ways. Just like in Set, the empty set is initial and the one-point set is terminal.
Example 1.3. The set R has topologies other than the usual metric topology. It has a
cofinite topology, a cocountable topology, and the sets [a, b) for a < b form a basis
for a topology on R called the lower limit topology (or the Sorgenfrey topology, or
the uphill topology, or the half-open topology). Unless specified otherwise, R will
be given the metric topology.
1
2
1 Examples and constructions
Example 1.4. In general, the intervals (a, b) = {x ∈ X : a < x < b} along with the
intervals (a, ∞) and (−∞, b) define a topology on any totally ordered set called the
order topology. The set R is totally ordered and the order topology on R coincides
with the usual topology.
Example 1.5. The integers Z are given the discrete topology unless specified otherwise. There’s another topology on Z for which sets
S(a, b) = {an + b : n ∈ N}
for a ∈ Z \ {0} and b ∈ Z, together with ∅, are open. Furstenberg [7] used this
topology in a delightful proof that there are infinitely many primes. One can check
that the sets S(a, b) are also closed in this topology. Since every integer except ±1
has a prime factor, it follows that
[
Z \ {−1, 1} =
S(p, 0).
primes p
Since the left hand side is not closed (no nonempty finite set can be open) there
must be infinitely many closed sets in the union on the right. Therefore, there are
infinitely many primes.
Example 1.6. Let R be a ring (commutative, with 1) and let spec(R) denote the set
of prime ideals of R. The Zariski topology on spec(R) is defined by declaring the
closed sets to be the sets of the form V (E) = {p ∈ spec(R) : E ⊆ p}, where E is
any subset of R.
Example 1.7. A norm on a real or complex vector space V is a function k k : V → R
(or C) satisfying
• kvk ≥ 0 for all vectors v with equality if and only if v = 0
• kv + wk ≤ kvk + kwk for all vectors v, w
• kαvk = |α|kvk for all scalars α and vectors v.
Every normed vector space is a metric space, hence topological space, with metric
defined by d(x, y) = k x − yk.qThe standard metric on Rn comes from the norm
Pn
2
defined by k(x 1 , . . . , x n )k =
i=1 |x i | . More generally, for any p ≥ 1, the pn
norm on R is defined by
1/p
n
X
k(x 1 , . . . , x n )k p := *
|x i | p +
, i=1
-
and the sup norm is defined by
k(x 1 , . . . , x n )k∞ := sup{|x 1 |, . . . , |x n |}.
These norms define different metrics, with different open balls, but for any of these
norms on Rn , the passage norm metric topology leads to the same topology. In
1.1 Examples and terminology
3
fact, for any choice of norm on a finite dimensional vector space, the corresponding
topological spaces are the same—not just homeormorphic, but literally the same.
Example 1.8. One can generalize from Rn to RN if one avoids sequences with diP
p
vergent norm. The set l p of sequences {x n } for which ∞
n=1 x n is finite is a subspace
of RN . Then l p with
1
∞
p
X
p
k{x i }k p := *
|x i | +
, i=1
is a normed vector space. It’s more difficitult to compare the topological spaces l p
for differenti p since the underlying sets are different. For instance, {1/n} is in l 2 but
not l 1 . Nonetheless, as topological spaces, the spaces l p are homeomorphic [11].
The the set l ∞ of bounded sequences with k{x i }k := sup |x i | is a also a normed
vector space, but it is not homeomorphic to l p for p , ∞.
1.1.2 Examples of continuous functions
The philosophy that objects are determined by their relationships with other objects
can be illustrated rather sharply in Top with the following example.
Example 1.9. Let S = {0, 1} with the topology {∅, {1}, S}—S is sometimes called the
Sierpinski two point space. Now, for any open set U ⊆ X, the characteristic function
χU : X → S defined by

 1 if x ∈ U,
χU (x) = 
 0 if x < U

is a continuous function, and every continuous function from X → S is of the
form χU for some open set U. Therefore, the open subsets of X are in one-to-one
correspondence with continuous functions X → S—the set hom(X, S) is a copy of
the topology of X.
Example 1.10. The reader seeing that the topology of a space X can be recovered
from hom(X, S) might wonder about whether the points can be recovered. But that’s
easy: since a point x ∈ X is the same as a map ∗ → X, the set of points of a space
X are isomorphic to the set hom(∗, X ).
A practical impact of the philosophy is that a space X can be studied by looking
at continuous functions either to or from a (usually simpler) space. For example,
the fundamental group of X involves functions from the circle S 1 to X. For another
example, sequences in a space X, which are the same as continuous functions from
the discrete space N to X, are used to probe topological properties of X. Maps from
X to {0, 1} detect connectedness, homotopy classes of maps ∗ → X reveal path
components, ...
Example 1.11. A path in a space X is a continuous function γ : [0, 1] → X. A loop
in a space X is a continuous function γ : [0, 1] → X with γ(0) = γ(1).
4
1 Examples and constructions
Example 1.12. If (X, d) is a metric space and x ∈ X, then the function f : X → R
defined by f (y) = d(x, y) is continuous.
Example 1.13. Unlike the categories Group and Vect where bijective morphisms
are isomorphisms, not every continuous bijection between topological spaces is a
homeomorphism. For example, the identity function id : (R, τdiscrete ) → (R, τusual )
is a continuous bijection that is not a homeomorphism. As we will see later, in the
subcategory of compact Hausdorff spaces, continuous bijections are always homeomorphisms.
1.2 The subspace topology
The subspace topology is often defined (for example, in [21]) as follows:
Definition 1.1. Let (X, τX ) be a topological space and let Y be any subset of X. The
subspace topology on Y is defined by {U ∩ Y : U ∈ τX }.
One checks that this definition does define a topology on Y . We’ll give two
characterizations of the subspace topology. The first one characterizes the subspace
topology as the coarsest topology on Y for which the inclusion map i : Y → X is
continuous. The second one is a universal property that characterizes the subspace
topology on Y by characterizing which functions into Y are continuous.
1.2.1 First characterization of the subspace topology
In order to describe the first characterization of the subspace topology, consider
a more general situation. Let (X, τX ) be a topological space and let S be any set
whatsoever. Consider a function
f : S → X.
It makes no sense to ask if f is continuous until S is equipped with a topology. There
always exist topologies on the set S that will make f is continuous—the discrete
topology is one. There is one topology, call it τf , that is the coarsest topology for
which τ is continuous. To see that such a topology exists, notice that the intersection
of any topologies on S for which f is continuous is again a topology on S for which
f is continuous. Therefore, the intersection of all topologies on S for which f is
continuous will be the coarsest topology for which f is continuous. Call it τf .
Note that τf has a simple explicit description as τf = { f −1 (U) : U ⊆ X is open}.
One sees that the subspace topology on a subset Y ⊆ X is the same as τi where
i : Y → X is the natural inclusion.
1.2 The subspace topology
5
Better Definition. Let (X, τX ) be a topological space and let Y be any subset of X.
The subspace topology on Y is the coarsest topology on Y for which the canonical
inclusion i : Y ,→ X is continuous.
Let X be a topological space, let S be any set, and let f : S → X be an injective
function. Then τf , the coarsest topology on S for which f is continuous, may be
called the subspace topology on S. This is a good definition, even though the set
S is not a subset of X. Since f is injective, the set S is isomorphic as a set to
its image f (S) ⊆ X; and the set S with the subspace topology τf determined by
f : S → X is homeomorphic to f (S) ⊆ X with the subspace topology determined
by the inclusion i : f (S) ,→ X. If f : S → X is not injective, then there is still a
coarsest topology τf on S that makes f continuous, but one doesn’t refer to it as the
subspace topology.
Definition 1.2. Suppose f : Y → X is a continuous injection between topological spaces. One calls f an embedding when the topology on Y is the same as the
subspace topology τf induced by f .
Example 1.14. Consider the set [0, 1] with the discrete topology. The map i :
([0, 1], τdiscrete ) → (R, τordinary ) is a continuous injection, but it is not an embedding.
The topology on the domain is not the subspace topology induced by i.
1.2.2 Second characterization of the subspace topology
Keeping in mind the philosophy that objects in a category are determined by morphisms to and from them, we might think about the subspace topology in two ways:
• The subspace topology τY determines the continuous maps to Y .
• The continuous maps to Y determine the subspace topology τY .
The second way of thinking about the subspace topology describes the important
universal property which characterizes precisely which functions into the subspace
are continuous—they are, reasonably, the functions Z → Y that are continuous
when they are regarded as functions into X.
Theorem 1.1. Let (X, τX ) be a topological space, let Y be a subset of X and let
i : Y ,→ X be the natural inclusion. The subspace topology on Y is characterized
by the following property:
Universal property for the subspace topology. For every topological space (Z, τZ )
and every function f : Z → Y , f is continuous if and only if i ◦ f : Z → X is
continuous.
Here’s a picture
6
1 Examples and constructions
X
i◦ f
i
Z
f
Y
Proof. Thinking of this theorem in two parts, we first verify that the subspace topology has the universal property. Second, we prove that any topology on Y that satisfies the universal property must be the subspace topology.
Let τY be the subspace topology on Y . Let (Z, τZ ) be any topological space and
let f : Z → Y . We have to prove that f : Z → Y is continuous if and only if
i ◦ f : Z → X is continuous. Suppose f is continuous, then i ◦ f : Z → X
is continuous since the composition of continuous functions is continuous. Now
suppose i ◦ f : Z → X is continuous. Let U be any open set in Y . Then U = i −1 (V )
for some open V ⊆ X. Since i ◦ f is continuous, the set (i ◦ f ) −1 (V ) ⊆ Z is open
in Z. Since (i ◦ f ) −1 (V ) = f −1 (U), we conclude that f −1 (U) is open. This proves
that f : Z → Y is continuous.
Now assume that τ 0 is a topology on Y and that τ 0 has the universal property.
We have to prove that this topology τ 0 equals the subspace topology τY . We are
assuming that when Y has the topology τ 0, then for every topological space (Z, τZ )
and for any function f : Z → Y , f is continuous if and only if i ◦ f is continuous. In
particular, if we let (Z, τZ ) be (Y, τY ) where τY is the subspace topology on Y , and
let f : Y → Y be the identity function, then we have the following picture
X
i ◦ idY = i
(Y, τY )
idY
i
(Y, τ 0 )
Since we know the function i ◦ idY = i : Y → X is continuous when Y has the
subspace topology τ, the universal property implies that idY : (Y, τY ) → (Y, τ 0 ) is
continuous. This implies that the subspace topology τY is finer than τ 0; i.e. τ 0 ⊆ τY .
To show that τY ⊆ τ 0, let (Z, τZ ) be (Y, τ 0 ) and let f = idY : (Y, τ 0 ) → (Y, τ 0 ). So
we have the following picture
X
i ◦ idY = i
(Y, τ 0 )
idY
i
(Y, τ 0 )
1.3 The quotient topology
7
Since idY is continuous, we must have i ◦ idY = i : Y → X continuous. That is, τ 0 is
a topology on Y for which the inclusion i : Y → X is continuous. Since the subspace
topology τY is the coarsest topology on Y for which i : Y → X is continuous, we
conclude that τY is coarser than τ 0; i.e., τY ⊆ τ 0 . The conclusion is that τ 0 = τY .
Example 1.15. In the subspace topology on Q ⊂ R, open sets are of the form Q ∩
(a, b) whenever a < b. Notice that the discrete and subspace topologies on Q are
not equivalent: for any rational r, the singleton set {r } is open in the former but not
in the latter.
1.3 The quotient topology
Let X be a topological space, let S be a set, and let π : X S be surjective. The
quotient topology on the set S is often defined as follows:
Definition 1.3. A set U ⊆ S is open in the quotient topology if and only if π −1 (U)
is open in X.
Before getting to the two characterizations of the quotient topology, let’s recall
how quotients of sets work. If ∼ is an equivalence relation on a set X, then X/∼
denotes the set of equivalence classes on X. If X is a set and π : X S is surjective then the set S is isomorphic to X/∼ where ∼ is the equivalence relation whose
equivalence classes are the fibers of π:
x ∼ y ⇔ π(x) = π(y).
The map π conveniently provides the isomorphism
'
S −→ X/∼
s 7→ π −1 (s).
Conversely, if ∼ is an equivalence relation on X, the natural projection π : X → X/∼
that sends x to its equivalence class defines a surjective function whose fibers are
are the equivalence classes of ∼.
So, one can always think of the quotient topology determined by a surjection
π : X S as being a topology defined on the set S or on the quotient of the
set X by the equivalence relation determined by the fibers of π, since S and X/∼
aren’t even distinguishable as sets. This is analogous to the two interpretations of
the subspace topology determined by an injection π : S ,→ X as being defined
defined on the set S, or on the subset π(S) ⊆ X.
8
1 Examples and constructions
1.3.1 The first characterization of the quotient topology
Notice that Definition 1.3 makes the quotient topology on S is the finest topology
for which the map π : X → S is continuous: saying U is open only if π −1 (U) makes
π continuous and saying U is open whenever π −1 (U) is open makes the quotient
topology the finest possible topology for which π is continuous. Thus, we have the
first characterization of the quotient topology, which leads to the better definition.
Better Definition. Let X be a topological space, let S be a set, and let π : X S be surjective. The quotient topology on S is the finest topology for which π is
continuous.
One word of caution: one should be careful when talking about the finest topology satisfying some property since such a topology may not exist. This is less of
an issue for the coarsest topology satisfying a property. The difference is that the
intersection of topologies is always a topology whereas the union of topologies is
usually not a topology.
1.3.2 The second characterization of the quotient topology
A topology on a set S is completely determined by saying what hom(S, Z ) is. So,
for a given surjection X S from a space to a set, a topology on S is determined
by specifying for every space Z, which functions S → Z are continuous. Given a
surjection π : X → S from a space X to a set S, the universal property characterizing
the quotient topology on the set S by specifying that the continuous maps S → Z
are precisely those whose precompositon with π are continuous functions X → Z.
Universal property for the quotient topology. Let X be a topological space, let S
be a set, and let π : X → S be surjective. For every topological space Z and every
function f : S → Z, f is continuous if and only if f ◦ π : X → Z is continuous.
Here is the picture:
X
f ◦π
π
S
f
Z
Theorem 1.2. The quotient topology is determined by the universal property described above.
Proof. Exercise.
The universal property of the quotient topology tells us precisely which functions
S → Z from a quotient to a space Z are continuous: they are continuous maps
f : X → Z that are constant on the fibers of π : X → S.
1.4 The product topology
9
Example 1.16. The map π : [0, 1] → S 1 defined by π(t) = (cos(2πt), sin(2πt)) is
a quotient map. Therefore, for any space Z, continuous functions S 1 → Z are the
same as continuous functions [0, 1] → Z which factor through π. That is, continuous
functions S 1 → Z are the same as paths γ : [0, 1] → X satisfying γ(0) = γ(1), the
loops in Z.
Example 1.17. The projective space RPn is defined to be the quotient of Rn+1 \ {0}
by the relation x ' λ x for λ ∈ R. So RPn is the set of lines through the origin in
Rn+1 and the quotient topology gives us the topology on this set of lines.
Example 1.18. Topological spaces are often defined by starting with a familiar space
and identifying points to obtain a quotient. For example, the picture below means
the space is obtained from the unit square I 2 in R2 with the opposite sides identified.
That is the topology on I 2 /∼ obtained from the map I 2 → I 2 /∼ where (x, 0) ∼ (x, 1)
and (0, y) ∼ (1, y).
The result is a space called the torus. We also have the Mobius band M and the
Klein bottle K, and the projective plane RP2 :
M
K
RP2
One should verify that the projective plane as defined by identifying opposite sides
of the square is homeomorphic to the definition given in Example 1.18.
1.4 The product topology
Let {Xα }α ∈ A be an arbitrary collection of topological spaces and consider the set
Y
X=
Xα .
α∈A
We’d like to make the set X into a topological space. Sometimes the product topology is defined by
10
1 Examples and constructions
Definition 1.4. The product topology on X is defined to be the topology generated
by the basis
Y





.
U
:
U
⊆
X
is
open,
and
all
but
finitely
many
U
=
X
α
α
α
α
α


α ∈ A

This definition, with its surprising “all but finitely many,” suggests that there are
better ways to define the product topology.
1.4.1 First characterization of the product topology
Again, we give two characterizations of the product topology. First, remember that
the set X comes with projection maps πα : X → Xα . Observe that there are topologies on X, such as the discrete topology, that make the projections πα : X → Xα
continuous. The intersection of all topologies that make the projections continuous
will be the coarsest topology for which the projections are continuous.
Better Definition. Let {Xα }α ∈ A be an arbitrary collection of topological spaces
Q
and let X = α ∈ A Xα . The product topology on X is defined to be the coarsest
topology on X for which all of the projections πα are continuous.
The proof that the better definition of the product topology is equivalent to the
Definition 1.4 is left as an exercise.
1.4.2 Second characterization of the product topology
The second characterization of the product topology amounts to saying precisely
which functions to the product are continuous. Let {Xα }α ∈ A be an arbitrary collecQ
tion of topological spaces and consider the set X = α ∈ A Xα . Keeping in mind
that the universal property of the product of sets says that set functions into X are
the same as collections of functions into the sets Xα , it’s not hard to guess that
Q
Z → X = α ∈ A Xα is continuous whenever all the components Z → X → Xα are
continuous.
Theorem 1.3. Let {Xα }α ∈ A be an arbitrary collection of topological spaces and
Q
let X = α ∈ A Xα . Let πα : X → Xα denote the natural projection. The product
topology on X is characterized by the following property.
Universal property for the product topology. For every topological space Z and every function f : Z → X, f is continuous if and only if for every α ∈ A, the component πα ◦ f : Z → Xα is continuous.
Here is the picture:
1.4 The product topology
11
X
f
Z
πα
Xα
fα
Proof. Exercise.
→
f
←
←
Example 1.19. Let X = R2 . One can write any function f : S → R2 in terms
of component functions f (s) = (x(s), y(s)). The components x(s) and y(s) are
simply the composition
S
←
x
R
y
2
←
π1
π2
→
→
→
→
←
R
R
The function f is continuous if and only if x and y are continuous. It’s good to realize that this way of specifying which functions into R2 are continuous completely
determines the topology on R2 .
Functions from R2 or more generally Rn can be a confusing, in part because our
familiarity with Rn can give unjustified topological importance to the maps R → R2
given by fixing one of the coordinates. Don’t make the mistake and think that a
function f : R2 → S is continuous if the maps x 7→ f (x, y0 ) and y 7→ f (x 0 , y) are
continuous for every x 0 and y0 , as in the picture below:
R
R
y7→ (x 0,←
y)
←
x7→ (x, y 0 )
→
←
→
←
←
R2
→
f
f (x, y 0 )
f (x 0, y)
S
→
→
For example, f : R2 → R defined by
xy



2 + y2

x
f (x, y) = 
0

if (x, y) , (0, 0)
if (x, y) = (0, 0)
is not continuous even though for any choice of x 0 or y0 , f (x, y0 ) and f (x 0 , y)
define continuous functions R → R.
12
1 Examples and constructions
1.5 The coproduct topology
Let {Xα } be a collection of topological spaces. We’d like to make the disjoint union
`
Xα into a topological space. As in the previous constructions, we’ll give two
characterizations of the coproduct topology.
1.5.1 The first characterization
`
Set theoretically, the disjoint union comes with functions Xα →
Xα and for a
first definition, we’d like a topology on X for which these inclusions are continuous.
`
There are many topologies on Xα for which these inclusions are continuous— the
`
indiscrete topology is one. The right topology to put on Xα is the finest topology
`
for which the maps Xα → Xα are continuous.
1.5.2 The second characterization
To characterize this topology a second way, remember that set theoretically, func`
tions from X are determined by collections of functions from Xα . Let X =
Xα
and let i α : Xα → X be the natural inclusions. The coproduct topology on X is
characterized by the following universal property.
Universal property for the disjoint union. For every topological space Z and every
function f : X → Z, f is continuous if and only if for every α ∈ A, f ◦ i α : Xα → Z
is continuous. Here is the picture:
→
X
←
Xα
←
←
iα
f
→
→
Z
f ◦i
α
Example 1.20. Any set X is the coproduct over its points of the one point set:
a
X'
{∗}
x ∈X
As spaces, however X `
x ∈X {∗}
if and only if X has the discrete topology.
1.7 Exercises
13
1.6 Homotopy and the homotopy category
Now that we have the product topology, we can talk about X × [0, 1] as a topological
space whenever X is a space. A homotopy from a map f : X → Y to a map g :
X → Y is a continuous function h : X × [0, 1] → Y satisfying h(x, 0) = f (x)
and h(x, 1) = g(x). Two maps f , g : X → Y are said to be homotopic if there is
a homotopy between them and we write f ' g. Homotopy defines an equivalence
relation on the maps hom(X,Y ) (check it), the equivalence classes of which are
called homotopy classes and are denoted [X,Y ]. One can check that the composition
of homotopic maps are homotopic, thus composition of homotopy classes of maps
is well defined. The homotopy category of topological spaces denoted hTop is the
category whose objects are topological spaces and whose morphisms are homotopy
classes of maps:
homh Top (X,Y ) := [X,Y ].
Two spaces are said to be homotopic if and only if they are isomorphic in hTop. That
is, X and Y are homotopic if and only if there exist maps f : X → Y and g : Y → X
so that g f ' id X and f g ' idY . Functors from hTop are called homotopy invariants
and functors from the category Top that descend to functors from hTop are called
homotopy functors.
Example 1.21. For an example, Rn ' ∗. Define f : ∗ → Rn by ∗ 7→ 0 and define
g : Rn → ∗ in the only way possible. Then g f = id∗ , and f g : Rn → 0 is a
map which is homotopic to idR2 , a homotopy being h(x,t) = t x. Spaces that are
homotopic to the point ∗ are said to be contractible.
Homotopy in topology is very important and will be discussed much more later,
but for now, it will be good to have the definition.
Often, a restricted notion of homotopy applies. For example, if α, β : [0, 1] → X
are paths from x to y, then a homotopy of paths is defined to be h : [0, 1] × [0, 1] → Y
satisfying h(t, 0) = α(t), h(t, 1) = β(t), and h(0, s) = x and h(1, s) = y for all
s,t ∈ [0, 1]. In other words, the homotopy fixes the endpoints of the path: for all s,
the path t 7→ h(t, s) is a path from x to y agreeing with α at s = 0 and β at s = 1.
1.7 Exercises
1.1. Draw a diagram of all the topologies on a three point set, indicating which are
contained in which.
1.2. In these notes Rn has been considered a topological space in two ways: as a
metric space with the usual distance function and as the product of n copies of R.
Prove that these are the same.
1.3. Check that the Zariski topology does in fact define a topology on spec(R) and
sketch a picture of spec(C[x]) and spec(Z). For a more challenging problem, sketch
a good picture of Z[x].
14
1 Examples and constructions
1.4. Give an example of a path p : [0, 1] → X connecting a to b in the space (X, τ)
where
X = {a, b, c, d}
and
τ = {∅, {a}, {c}, {a, c}, {a, b, c}, {a, d, c}, X }.
1.5. Prove that any two norms on a finite dimensional vectors space (over R or C)
give rise to homeomorphic topological spaces.
1.6. Prove that l ∞ is not homeomorphic to l p for p , ∞.
1.7. Let C([0, 1]) denote the set of continuous functions on [0, 1]. The following
define norms on C([0, 1]):
k f k∞ = sup | f (x)|.
x ∈[0, 1]
k f k1 =
1
Z
| f |.
0
Prove that the topologies on C([0, 1]) coming from these two norms are different.
Q
1.8. Prove Theorem 1.3. That is, prove that X = α ∈ A Xα with the product topology has the universal property. Then, prove that if X is equipped with any topology
having the universal property, then that topology must be the product topology.
1.9. Are the subspace and product topologies are consistent with each other? Let
{Xα }α ∈ A be a collection of topological spaces and let {Yα } be a collection of subsets;
Q
each Yα ⊆ Xα . There are two things you can do to put a topology on Y = α ∈ A Yα :
1. You can take the subspace topology on each Yα , then form the product topology
on Y .
2. You can take the product topology on X, view Y as a subset of X and equip it
with the subspace topology.
Is the outcome the same either way? If yes, prove it using only the universal properties. If no, give a counterexample.
1.10. Prove that the quotient topology is characterized by the universal property
given in Section 1.3.
1.11. Are the quotient and product topologies are compatible with each other? Let
{Xα }α ∈Λ be a collection of topological spaces, let {Yα }α ∈Λ be a collection of sets,
Q
and let {πα : Xα Yα }α ∈Λ be a collection of surjections. Let X = α Xα and
notice that you have a surjection π : X Y . There are two ways to put a topology
Q
on Y = α ∈ A Yα :
1. Take the quotient topology on each Yα , then form the product topology on Y .
2. Take the product topology on X, then put the quotient topology on Y .
Is the outcome the same either way? If yes, prove it using only the universal properties. If no, give a counterexample.
1.7 Exercises
15
1.12. Suppose that X is a topological space and f : X → S is surjective. Define an
equivalence relation on X by x ∼ x 0 ⇔ f (x) = f (x 0 ). Let
R = {(x, x 0 ) ∈ X × X : f (x) = f (x 0 )}
One has two maps, call them r 1 : R → X and r 2 : R → X defined by the composition with the two natural projections X × X → X.
ri
π1
→ X×X
←
←
←
R
→ X
π 2→
Learn what a coequalizer is and prove that the set S with the quotient topology is
the coequalizer of r 1 and r 2 .
1.13. Definition 1.5. Let X and Y be topological spaces. A function f : X → Y is
called open (or closed) if and only if f (U) is open (or closed) in Y whenever U is
open (or closed) in X.
Let (X, τX ) and (Y, τY ) be topological spaces and suppose f : X → Y is a continuous surjection.
(a) Give an example to show that f may be open but not closed.
(b) Give an example to show that f may be closed but not open.
(c) Prove that if f is either open or closed, then the topology τY on Y is equal to τf ,
the quotient topology on Y .
1.14. Consider the closed disk D 2 and the two sphere S 2 :
D 2 = {(x, y) ∈ R2 : x 2 + y 2 ≤ 1}
S 2 = {(x, y, z) ∈ R3 : x 2 + y 2 + z 2 = 1}.
Consider the equivalence relation on D 2 defined by identifying every point on S 1 ⊆
D 2 . So each point in D 2 \ S 1 is a one point equivalence class, and the entire ∂(D)
is one equivalence class. Prove that the quotient D/ ∼ with the quotient topology is
homeomorphic to S 2 .
Chapter 2
Connectedness and compactness
One of the classical aims of topology is to classify topological spaces by their topological
type, or in other terms to find a complete set of topological invariants.
— Samuel Eilenberg
In this chapter, we discuss a few important topological properties: connected,
Hausdorff, and compact.
2.1 Connectedness
This section contains the main ideas about connectedness. The definitions are collected up front and the main results follow. The proofs are mostly left as exercises,
but the reader can find them in most any classic text on topology [28, 21, 12, 15].
2.1.1 Definitions, theorems, and examples
Definition 2.1. A topological space X is connected if and only if one of the following equivalent conditions holds:
1. X cannot be expressed as the union of two disjoint nonempty open sets.
2. Every continuous function f : X → {0, 1} is constant.
Notice that {0, 1} could be replaced by any two point set with the discrete topology
in the second condition.
There’s a second, probably more important kind of connectedness. Recall that a
path from x to y in a topological space X is a map γ : I → X with γ(0) = x and
γ(1) = y.
Definition 2.2. A topological space X is said to be path connected if and only if for
all x, y ∈ X there is a path that connects x and y.
17
18
2 Connectedness and compactness
There is an equivalence relation defined by x ∼ y if and only if there is a path in
X connecting x and y. The constant path shows ∼ is reflexive. If f is a path from x
to y, then g defined by g(t) = f (1−t) is a path from y to x, showing ∼ is symmetric.
To see that ∼ is transitive, it’s helpful to define composition of paths: If f is a path
from x to y and g is a path from y to z define the product g · f to be the path from
x to z obtained by first traversing f from x to y and then traversing g from y to z,
each at twice the speed:

 f (2t),
0 ≤ t ≤ 1/2
(g · f )(t) = 
 g(2t − 1), 1/2 ≤ t ≤ 1.

The equivalence classes of ∼ are called the path components of X and the set of all
path components is denoted by π0 (X ).
Notice that the path components of X are homotopy classes of maps ∗ → X: a
point x ∈ X is a map ∗ → X and a path between two points ∗ → X is a homotopy
between the maps. There’s also an equivalence relation on X defined by x ∼0 y
if and only if there’s a connected subspace of X that contains both x and y. The
equivalence classes of ∼0 are called the connected components of X, but we don’t
use a special symbol to denote them.
And now we highlight some of the theorems.
Theorem 2.1. If X is connected (or path connected) and f : X → Y then f (X ) is
connected (or path connected).
Proof. Exercise.
Corollary 2.1. Connected and path connected are topological properties. Furthermore, being connected or path connected is a homotopy invariant.
Theorem 2.2. Let X be a space and f : X → Y be a surjective map. If Y is connected in the quotient topology and each fiber f −1 (y) is connected, then X is connected.
Proof. Let g : X → {0, 1}. Since the fibers of f are connected, g must be constant
on the fibers of f . Therefore g factors through f : X → Y and one gets a map
g : Y → {0, 1} that fits into this diagram.
←
X
←
Y
←
→
f
g
g
→
→ {0, 1}
Since Y is connected, g is constant, hence the composition g = g f is constant.
Theorem 2.3. Suppose X = ∪α ∈ A Xα and that Xα is connected (or path connected)
for each α ∈ A. If there is a point x ∈ ∩α ∈ A Xα then X is connected (or path
connected).
2.1 Connectedness
19
Proof. Exercise.
Theorems 2.3 and 2.2 are typical in a certain way. Theorem 2.3 involves a space
decomposed into a collection of open sets. Knowing something about each open
set (that they’re connected) and knowing something about the intersection (it’s
nonempty) tells you something about the whole space (it’s connected). Theorem
2.2 involves a space X decomposed into fibers over a base space. Here, you know
something about the base space (it’s connected) and about the fibers (they’re connected) and you can conclude something about the total space (it’s connected). This
approach to extending knowledge about parts to knowledge of the whole appears
over and over again in mathematics.
(
√ )
Example 2.1. The rational numbers Q are not connected since x ∈ Q : x < 2
(
√ )
and x ∈ Q : x > 2 are two nonempty open subsets of rationals whose union is
Q. In fact, the rationals are totally disconnected meaning that the only connected
subsets are singletons.
Theorem 2.4. The connected subspaces of R are intervals.
Proof. Suppose A is a connected subspace of R which is not an interval. Then there
exist x, y ∈ A such that x < z < y for some z < A. Thus A = ( A ∩ (−∞, z)) ∪ ( A ∩
(z, ∞)) is a separation of A into two disjoint nonempty open sets.
Conversely suppose I is an interval with I = U ∪V where U and V are nonempty,
open and disjoint. Then there exist x ∈ U and y ∈ V , and we may assume x < y.
Since the set U 0 = [x, y) ∩ U is nonempty and bounded above, s = sup U 0 exists by
the completeness of R. Moreover, since x < s ≤ y and I is an interval, either s ∈ U
or s ∈ V and so (s − δ, s + δ) ⊆ U or (s − δ, s + δ) ⊆ V for some δ > 0. If the former
holds, then s fails to be an upper bound on U 0. If the latter, then s − δ is an upper
bound for U 0 which is smaller than s. Both lead to a contradiction.
The fact that the interval I = [0, 1] is connected proves the following:
Theorem 2.5. Path connected implies connected.
The next example is in fact too nice to be labeled example—we’ll call it a theorem.
Theorem 2.6. Every continuous function f : [−1, 1] → [−1, 1] has a fixed point.
Proof. Suppose f : [−1, 1] → [−1, 1] is a continuous function for which f (x) , x
for all x ∈ [−1, 1]. In particular we have f (−1) > −1 and f (1) < 1. Now define a
map g : [−1, 1] → {−1, 1} by
g(x) =
x − f (x)
.
|x − f (x)|
Then g is continuous and g(−1) = −1 and g(1) = 1. But this is impossible since
[−1, 1] is connected.
20
2 Connectedness and compactness
We’ve just proved the n = 1 version of Brouwer’s Fixed Point Theorem which
states that any continuous function D n → D n must have a fixed point. The result
when n = 2, is proved in Chapter ??.
Here’s another nice result that follows from the connectedness of intervals [22,
29].
Theorem 2.7. Every convex polygon can be partitioned into two convex polygons,
each having the same area and same perimeter.
Proof. Let P be a convex polygon and first note that finding a line which bisects the
area of P is not difficult. Simply take a vertical line and consider the difference of the
area on the left and the right. As the line moves from left to right the difference goes
from negative to positive continuously and therefore must be zero at some point.
Now, there was nothing special about the line being vertical. There’s a line in
every direction which bisects P. So now, start with the vertical line and consider the
difference between the perimeter on the left and the perimeter on the right. Rotate
this line in such a way that it always bisects the area of P and observe that the difference between the perimeters switches sign as you get halfway around. Therefore,
there exists a line which cuts P into two convex polygons, both with equal areas and
equal perimeters.
2.1.2 The functor π0
The assignment X 7→ π0 (X ) is the object part of a functor π0 : Top → Set. If f :
X → Y is continuous and A is a path component of X, f ( A) is necessarily connected
hence contained in a unique path component of Y . So the function π0 f which sends
A to the path component containing f ( A) defines a function from π0 (X ) → π0 (Y ).
To summarize the functor,
π0 : Top → Set
X→
7 π0 (X )
f
π0 f
X −→ Y →
7
π0 (X ) −→ π0 (Y )
The fact that functors respect composition when applied to morphisms makes
them quite powerful. For example, let’s recast the proof of Theorem 2.6 using the
functor π0 . Suppose f : [−1, 1] → [−1, 1] is continuous. If f (x) , x for any x, then
the map g : [−1, 1] → {−1, 1} defined by
g(x) =

 −1
x − f (x)
=
|x − f (x)|  1

is continuous. So, we have the diagram of spaces
if x < f (x),
if x > f (x).
2.1 Connectedness
21
id
←
←
{−1, 1} -←i → [−1, 1]
→
{−1, 1}
→
g
Apply π0 to get a diagram of sets
π 0 id=id
←
0
←
{−1, 1} -←π →
{∗}
i
→
π 0 g→ {−1, 1}
which is impossible—no map {−1, 1} → {∗} can be left invertible and no map {∗} →
{−1, 1} can be right invertible.
2.1.3 Constructions and connectedness
When considering a particular topological property, it’s a good idea to determine
if it’s preserved by particular constructions. In general, there’s no reason to expect
a property to be preserved by subspaces, products, coproducts, quotients, etc... For
connectedness, it takes no imagination to see that subspaces of connected spaces
are not necessarily connected. The situation for quotients and products, on the other
hand, is as good as it gets. Since quotient maps are continuous surjections, Theorem
2.1 immediately implies that quotients of connected (or path connected) spaces are
connected (or path connected). Products of connected spaces are also connected:
Theorem 2.8. Let {Xα }α ∈ A be a collection of connected (or path connected) topoQ
logical spaces. Then X = α ∈ A Xα is connected (or path connected).
Proof. We’ll prove it for path connected and leave the connected part as an exercise.
Suppose that Xα is path connected for every α ∈ A. Let a, b ∈ X. Since each Xα
is path connected, there exists a path pα : [0, 1] → Xα connecting aα to bα . The
universal property of the product topology implies that the function p : [0, 1] → X
uniquely defined by πα ◦ p = pα is a continuous function p : [0, 1] → X. Then p is
a path from a to b.
Now, the situation for unions benefits from some elaboration. Theorem 2.3 involves unions of connected spaces that are not disjoint. Of more categorical interest
are disjoint unions; i.e., the coproducts in Top. The categorically minded reader
may consult the entry on connectedness at nLab [1]. Since the connected components {Xα } of a space X partition the space X, every topological space X as a set is
the disjoint union of its connected components
a
X=
Xα .
However, whether X, as a topological space, is the coproduct of its connected components depends on whether the components Xα are open. For example, the con-
22
2 Connectedness and compactness
`
nected components of the rationals Q consist of singletons {r }. As a set, Q = {r }
`
but certainly not as a topological space: r ∈Q {r } is just the countable discrete topological space.
The idea that a space is connected if and only if the only maps from it to a
two point discrete space can made more categorical. For any space X, there is exactly one function X → ∗. Now, think of a two point discrete space as the co`
`
product ∗ ∗. If X is connected, there are precisely two functions X → ∗ ∗.
Namely, the two constant functions, X maps to the first point and X maps to the
`
second point. So the set hom(X, ∗ ∗) is the two point set, canoncially isomorphic
`
to hom(X, ∗) hom(X, ∗). On the other hand if X is not connected there are more
`
than two maps hom(X, ∗ ∗). For example, if X = [0, 1]∪[2, 3], there are four func`
`
`
tions X → ∗ ∗. So, the set hom(X, ∗ ∗) is not equal to hom(X, ∗) hom(X, ∗).
This leads to a definition of connectedness that makes sense in any category that has
coproducts:
Theorem 2.9. A space X is connected if and only if the functor h X := hom(X, −)
preserves coproducts.
In summary, path-connected and connected are preserved by products and quotients, but not by subspaces or coproducts.
2.1.4 Local (path) connectedness
Many topological properties have local versions.
Definition 2.3. A topological space is locally connected (or locally path connected)
if for each x ∈ X and every neighborhood U ⊆ X of x, there is a connected (or path
connected) neighborhood V of x with V ⊆ U.
Example 2.2. Let X denote the graph of f (x) = sin(1/x) where x > 0 along with
part of the y-axis ranging from (0, −1) to (0, 1). The space X, called the topologist’s
sine curve, is connected but not path connected.
If X is locally connected, then the connected components are open. This has
several consequences, including the following theorem.
Theorem 2.10. In any locally path connected topological space, the connected
components and path components are the same.
Proof. Exercise.
Example 2.3. The topologist’s sine curve from Example 2.2, then, is connected but
not locally connected. The space [0, 1]∪[2, 3] is locally connected but not connected.
The above illustrate that neither connectedness nor local connectedness implies
the other, and the same is true if we replace “connected" with “path connected."
2.2 Hausdorff spaces
23
Example 2.4. Let C = { n1 : n ∈ N} ∪ 0 and set X = (C × [0, 1]) ∪ ([0, 1] × {0}).
Then X, called the comb space, is path connected but not locally path connected.
On the other hand, the set [0, 1] ∪ [2, 3] in R with the subspace topology is locally
path connected but not path connected.
1
0
1
2.2 Hausdorff spaces
Now is a good time to define a topological property called Hausdorff.
Definition 2.4. A space X is Hausdorff if and only if for every two points x and y,
there exist disjoint open sets U and V with x ∈ U and y ∈ V .
First, one should check that Hausdorff defines a topological property, but not a
homotopy invariant property. Then, one should look at which constructions preserve
the Hausdorff property. One finds that subspaces of Hausdorff spaces are Hausdorff,
products of Hausdorff spaces are Hausdorff, coproducts of Hausdorff spaces are
Hausdorff, but quotients of Hausdorff spaces are not-necessarily Hausdorff. In fact,
quotients of Hausdorff spaces are a great source of non-Hausdorff spaces throughout
the mathematical world.
Theorem 2.11. Every space X is the quotient of a Hausdorff space H.
Proof. Omitted. See [23].
Example 2.5. Metric spaces are Hausdorff. To see this, let x and
y be points
in a
metric space. If x , y, then d := d(x, y) > 0. Then B x, d2 and B y, d2 are
disjoint open sets separating x and y.
Theorem 2.12. A space X is Hausdorff if and only the diagonal map ∆ : X → X × X
is closed.
Proof. Exercise.
The Hausdorff property interacts with other topological properties (like compactness) is some interesting ways.
24
2 Connectedness and compactness
2.3 Compactness
2.3.1 Definitions, theorems and examples
Let’s start with the definition and the fact that compactness defines a topological
property:
Definition 2.5. A collection U of open subsets of a space X is an open cover for X
if the union of sets in U contains X. The space X is compact if and only if every
open cover of X has a finite subcover.
Theorem 2.13. If X is compact and f : X → Y then f (X ) is compact.
Proof. Exercise.
Corollary 2.2. Compactness is a topological property.
One way to think of compact spaces is that they are somehow small—not in terms
of cardinality, but in terms of roominess. For example, if you squeeze an infinite set
of points into the unit interval, they get cramped—for any > 0, there are two points
that are less than apart. But, it’s easy to fit an infinite number of points in the real
line so that they’re all spread out. In general, a point x is called a limit point of a set
X if every neighborhood of x contains a point of X \ {x}. This idea is summarized
in the following simple theorem.
The Bolzano-Weierstrass Theorem. Every infinite set in a compact space has a
limit point.
Proof. Suppose that F is an infinite subset with no limit points. If x not a limit point
of F and x < F, there is an open set Ux around x that misses F. If x is not a limit
point of F and x ∈ F, then there is an open set Ux with Ux ∩ F = {x}. Then {Ux } x ∈X
is an open cover of X. Notice that there can be no finite subcover Ux1 , . . . ,Ux n since
(Ux1 ∪ · · · ∪ Ux n ) ∩ F = {x 1 , . . . , x n }, and cannot contain the infinite set F.
Example 2.6. There are noncompact spaces for which every infinite subset has a
limit point. For instance, take R with topology {(x, ∞) : x ∈ R}, together with ∅ and
R. This space is not compact, but any set (infinite or not) has a limit point (infinitely
many, in fact).
Definition 2.6. Let S be a collection of sets. We say that the collection S has the
finite intersection property if and only if for every finite subcollection A1 , . . . , An ⊂
S, the intersection A1 ∩ · · · ∩ An , ∅. We abbreviate the finite intersection property
by FIP.
Theorem 2.14. A space X is compact if and only if every collection of closed subsets
of X with the FIP has nonempty intersection.
Proof. Exercise.
2.3 Compactness
25
Theorem 2.15. Closed subsets of compact spaces are compact.
Proof. Let X be compact with C ⊆ X closed and suppose U = {Uα }α ∈ A is an open
cover of C. Then X r C together with U forms an open cover of X. Since X is
n in U , possibly together with X r C,
compact, there are finitely many sets {Ui }i=
1
n
which covers X. Thus {Ui }i=1 is a finite subcover for C.
Compact subsets of Hausdorff spaces are quite nice—they can be separated from
points by open sets.
Theorem 2.16. Let X be Hausdorff. For any point x ∈ X and any compact set
K ⊂ X \ {x} there exist disjoint open sets U and V with x ∈ U and K ⊂ V .
Proof. Let x ∈ X and let K ( X be compact. For each y ∈ K, there are disjoint
open sets Uy and Vy with x ∈ Uy and y ∈ Vy . The collection {Vy } is an open cover
of K, hence there is a finite subcover {V1 , . . . ,Vn }. Let U = U1 ∩ · · · ∩ Un and
V = V1 ∪ · · · ∪ Vn . Then U and V are disjoint open sets with x ∈ U and K ⊂ V .
Corollary 2.3. Compact subsets of Hausdorff spaces are closed.
Corollary 2.4. If X is compact and Y is Hausdorff then every map f : X → Y is
closed. In particular, if f is injective, then it is an embedding, if f is surjective, it is
a quotient map, and if f is bijective, it is a homeomorphism.
Proof. Let f : X → Y be a map from a compact space to a Hausdorff space and let
C ⊂ X be closed. Then C is compact, so f (C) is compact, so f (C) is closed.
2.3.2 Constructions and compactness
The reader will realize that we’ve proved that quotients of compact spaces are
compact. Subspaces of compact spaces are not compact in general, but Theorem
2.15 says closed subspaces of compact spaces are compact. Coproducts of compact
spaces are not compact—just look at the coproduct of infinitely many copies of
the point. There are a few interesting things to explain regarding compactness and
products.
First, we have Tychonoff’s theorem and some of its corollaries.
Tychonoff’s Theorem. The product of compact spaces is compact.
Proof. In Chapter 3.4.
Corollary 2.5. A subset of Rn is compact if and only if it is closed and bounded.
Proof. Suppose that K ⊂ Rn is compact. Since the cover of K consisting of open
balls centered at the origin of all possible radii must have a finite subcover, K must
be bounded. Since Rn is Hausdorff and all compact subsets of a Hausdorff space
must be closed, K is closed.
26
2 Connectedness and compactness
Conversely (and this is the part that uses the Tychonoff theorem), suppose that
K ⊂ Rn is closed and bounded. Since K is bounded, the projection of K onto the
i − th coordinate is bounded; i.e., πi (K ) ⊂ [ai , bi ] for each i. Then K ⊂ [a1 , b1 ] ×
[a2 , b2 ] × · · · × [an , bn ]. Since each set [ai , bi ] is compact, the Tychonoff theorem
implies that the product [a1 , b1 ] × [a2 , b2 ] × · · · × [an , bn ] is compact. Since any
closed subset of a compact space is compact, we conclude that K is compact.
Corollary 2.6. Continuous functions from compact spaces to R have both a global
maximum and a global minimum.
The characterization of compact subsets of Rn as closed and bounded may be
familiar from analysis, but take note that bounded is not a topological property. The
space R is not a bounded metric space, (0, 1) is a bounded metric space and as
topological spaces R (0, 1).
Example 2.7. Like any finite set, the one point set ∗ is compact. Since R is not
compact but is homotopy equivalent to ∗, we see that compactness is not a homotopy
invariant.
Finally, we have the so-called tube lemma, which isn’t a corollary of Tychonoff,
but concerns compact sets and products. First, an example:
Example 2.8. Take for example the open set
U := {(x, y) ⊂ R2 : 0 < x < 1, 0 < y < x}.
Here U is just the( interior
of the triangle with corners
)
(0,
0), (1, 0), and (1, 1). Con
sider the set A × 12 where A is the interval A = 12 , 1 . Then A × 12 − , 21 + is
not contained in U for any > 0. But if A were compact,...
The Tube Lemma. For any open set U ⊂ X ×Y and any set K ×{y} ⊂ U with K ⊂ X
compact, there exist open sets V ⊂ X and W ⊂ Y with K × {y} ⊂ V × W ⊂ U.
Proof. For each point (x, y) ∈ K × {y}, there is arex open sets Vx ⊂ X and W x ⊂ Y
with (x, y) ⊂ Vx × W x ⊂ U. Then, {Vx } x ∈K is an open cover of K; take a finite
subcover {V1 , . . . ,Vn }. Then V = V1 ∪ · · · ∪ Vn and W = W1 ∩ · · · ∩ Wn are open
sets with K × {y} ⊂ V × W ⊂ U.
2.3.3 Local compactness
There’s a local version of compactness.
Definition 2.7. A space X is locally compact if and only if for every point x ∈ X
there exists a compact set K and an open set U with x ∈ U ⊂ K.
2.3 Compactness
27
Example 2.9. Every compact space is locally compact. Rn is locally compact. Every
discrete space is locally compact. The real line with the lower limit topology τll (see
Example 1.3) is not locally compact.
Now, the image of a locally compact space need not be locally compact. For example, id : (R, τdiscrete ) → (R, τll ) gives an example. Nonetheless, locally compact
is a topological property.
Now, the product and quotient topologies are not compatible in the sense of Problem 1.11. The hypothesis of locally compact and Hausdorff makes the situation
much better.
Theorem 2.17. If X1 Y1 and X2 Y2 are quotient maps and Y1 and X2 are
locally compact and Hausdorff, then X1 × X2 Y1 × Y2 is a quotient map.
Proof. Exercise.
Exercises
2.1. Prove that the two items in Definition 2.1 are indeed equivalent.
2.2. Define an arc in a space X to be an path I → X that is an embedding. A
space X is arc connected if every two points can be connected by an arc. Prove that
arc-connected ⇒ path-connected ⇒ connected. Give examples to show that the
implications are strict.
2.3. A map X → Y from a space X is locally constant if for each x ∈ X there is an
open set U with x ∈ U and f |U constant. Prove or disprove: if X is connected and
Y is any space, then every locally constant map f : X → Y is constant.
2.4. Show that every countable metric space M with at least two points must be
disconnected. Construct a topological space with more than two elements which is
both countable and connected.
2.5. In a variation of the topology on Z in Example 1.5, consider the natural numbers
N with topology with a basis consisting of
{ak + b : k ∈ N and a, b ∈ N are relatively prime}
Prove that N with this topology is connected. [8]
Q
Q
2.6. Let {Xα } be a collection of spaces. Prove that π0 ( Xα ) π0 (Xα ). The
special case π0 (Xα ) = ∗ for all α is the statement that the product of path connected
spaces is path connected.
2.7. Prove that a space X is connected if and only if the functor h X = hom(X, −)
preserves coproducts.
28
2 Connectedness and compactness
2.8. Show that Q ⊆ R with the subspace topology is not locally compact.
2.9. Define a space X to be pseudocompact iff every real valued function on X is
bounded. Prove that if X is compact then X is pseudocompact and give an example
of a pseudocompact space that is not compact.
2.10. Give examples showing that locally compact is not preserved by subspaces,
quotients, or products.
2.11. Let U be an open cover of a compact metric space X. Show that there exists
an > 0 such that for every x ∈ X, B(x, ) is contained in some U ∈ U. Such an is called a Lebesgue number for U .
2.12. Show that Z endowed with the arithmetic progression topology of Example
1.5 is not locally compact.
2.13. Suppose (X, d) is a compact metric space and f : X → X is an isometry, i.e.
for all x, y ∈ X, d(x, y) = d( f (x), f (y)). Prove f is a homeomorphism.
2.14. Let X be a space and suppose A, B ⊆ X are compact. Prove or disprove:
(a) A ∩ B is compact.
(b) A ∪ B is compact.
If a statement is false, find a sufficient condition on X which will cause it to be true.
(
)
P
2
2
2.15. Let B = {x n } ∈ l 2 : ∞
n=1 x n ≤ 1 be the closed unit ball in l . Show that B
is not compact.
2.16. Prove that if Y is compact then for any space X the projection X × Y → X is a
closed map. Give an example of spaces X and Y for which the projection X ×Y → X
is not closed.
2.17. Show that the product of Hausdorff spaces is Hausdorff. Give an example to
show that the quotient of a Hausdorff space need not be Hausdorff.
2.18. For any map f : X → Y , the set Γ = {(x, y) ∈ X × Y : y = f (x)} is called the
graph of f . Suppose now that X is any space and Y is compact Hausdorff. Prove that
Γ is closed if and only if f is continuous. Is the compactness condition necessary?
(This is called the closed graph theorem.)
2.19. Let X be a Hausdorff space with f : X → Y a continuous closed surjection
such that f −1 (y) is compact for each y ∈ Y . Prove that Y is Hausdorff.
2.20. Prove or disprove: If f : X → Y is a continuous bijection and X is Hausdorff
then Y must be Hausdorff.
2.21. Suppose A is a subspace of X. We say a map f : A → Y can be extended to X
if there is a continuous map g : X → Y with g = f on A.
2.3 Compactness
29
(a) Prove that if A is dense in X and Y is Hausdorff, then f can be extended to X in
at most one way.
(b) Give an example of spaces X and Y , a dense subset A, and a map f : A → Y
such that f can be extended to X in more than one way.
(c) Give an example of spaces X and Y , a dense subset A, and a map f : A → Y
such that f cannot be extended.
2.22. Prove or disprove: X is Hausdorff if and only if
{(x, x, x, x, x, . . .) ∈ X N : x ∈ X }
is closed in X N .
)
(
2.23. Let K = n1 : n ∈ N ⊂ R. The K topology on R is the topology generated by
the basic open sets (a, b) and (a, b) \ K where a < b.
(a) Prove that [0, 1] is not a compact subspace of R with the K-topology.
(b) More generally, prove that if τ is any topology on [0, 1] finer than the ordinary
one, then [0, 1] cannot be compact in the topology τ.
(c) Prove that if τ is any topology on [0, 1] coarser than the usual one, then [0, 1]
cannot be Hausdorff in the topology τ.
Chapter 3
Limits of sequences and nets
Given any point of S, there is a sphere of positive radius having this point as center, all of
whose points belong to S. More generally, I call any set of points possessing this property
an open domain of n dimensions.
— Karl Theodor Wilhelm Weierstrass, 1865
In this chapter we make a slight departure from the categorical point of view.
We’ll spend a little time inside of spaces to discuss some ideas that may be familiar
from analysis.
3.1 Closure and interior
Every subset B of a topological space X has a closure B which is the smallest closed
set containing B and an interior B◦ which is the largest openset contained in B. A
◦
set B is called dense if B = X and is called nowhere dense if B = ∅.
Recall that a point x is a limit point of a set B if every open set around x contains
a point of B \ {x}. The closure B consists of B together with all of its limit points.
A point x is a boundary point of B if every open set containing x contains a point in
B and a point in the complement of B.
3.2 Sequences
Definition 3.1. Let X be a topological space. A sequence in X is a function x : N →
X. We usually write x n for x(n) and may denote the sequence {x n }. A sequence {x n }
converges to z ∈ X if and only if for every open set U containing z, there exists an
N ∈ N so that if n ≥ N, x n ∈ U. If {x n } converges to z ∈ X we write {x n } → z. A
subsequence of a sequence x is the composition x ◦ k of x where k : N → N is an
31
32
3 Limits of sequences and nets
increasing injection. One often writes x k i for x(k (i)) and dentoes the subsequence
by {x k i }.
Here are a few examples:
Example 3.1. Let A = {1, 2, 3} with the topology τ = {∅, {1}, {1, 2}, A}. Then the
constant sequence 1, 1, 1, 1, . . . converges to 1; it also converges to 2 and to 3.
Example 3.2. Consider Z with the cofinite topology. For any m ∈ Z, the constant
sequence m, m, m, . . . converges to m and only to m. For if l , m, the set R \ m is an
open set around l contiaining no elements of the sequence.
The sequence {n} = 1, 2, 3, 4, . . . converges to m for every m ∈ Z. To see this, let
m be any integer and let U be a neighborhood of m. Since Z \ U is finite, there can
only be finitely many natural numbers in Z \ U. Let N be larger than the greatest
natural number in Z \ U. Then, if n > N, n ∈ U, proving that {n} → m.
Example 3.3. Consider R with the usual topology. If {x n } → x, then {x n } does not
converge to any number y , x. To prove it, we can find disjoint open sets U and
V with x ∈ U and y ∈ V (we can be explicit if necessary: U = (x − c, x + c) and
V = (y − c, y + c) where c = 21 |x − y|). Then, there is a number N so that x n ∈ U
for all n ≥ N. Since U ∩ V = ∅, V cannot contain any x n for n ≥ N and hence {x n }
does not converge to y.
Sequences can be used to detect certain properties of spaces, subsets of spaces,
and functions between spaces. Before continuing, let’s introduce a couple more
topological properties—two of the so-called “separation” axioms.
Definition 3.2. We say
• A topological space X is T0 iff for every pair of points x, y ∈ X there exists an
open set containing one, but not both of them.
• A topological space X is T1 iff for every pair of points x, y ∈ X there exists open
sets U and V with x ∈ U, y ∈ V with x < V and y < U.
Observe that T0 and T1 define topological properties. A space X with the property
that for every pair of points x, y ∈ X there exist open sets U and V with x ∈ U,
y ∈ V with U ∩ V = ∅ is sometimes called T2 , but we’ve already named the property
Hausdorff, after Felix Hausdorff who originally used the axiom in his definition of
“neighborhood spaces." [10].
Here are a few theorems about sequences. Keep the examples above in mind.
Theorem 3.1. A space X is T1 if and only for any x ∈ X, the constant sequence
x, x, x, . . . converges to x and only to x.
Proof. Suppose X is T1 and x ∈ X. It’s clear that x, x, x, . . . → x. Let y , x. Then
there exists an open set U with y ∈ U and x < U. Therefore, x, x, x, . . . cannot
converge to y.
For the converse, suppose that X is not T1 . Then, there exist two distinct points x
and y for which every open set around y contains x. The sequence x, x, x, . . . → y.
3.2 Sequences
33
Theorem 3.2. If X is Hausdorff, then sequences in X have at most one limit.
Proof. Let X be Hausdorff, let {x n } be a sequence with {x n } → x and {x n } → y.
If x , y, then there are disjoint open sets U and V with x ∈ U and y ∈ V. Since
{x n } → x there is a number N so that x n ∈ U for all n ≥ N. Since {x n } → y there
is a number K so that x n ∈ U for all n ≥ K. Let M = max{N, K }. Since M ≥ N
and M ≥ K we have x M ∈ U and x M ∈ V contradicting the fact that U and V are
disjoint.
Theorem 3.3. If f : X → Y is continuous then for all sequences {x n } → x in X,
the sequence { f (x n )} → f (x).
Proof. Exercise.
Theorem 3.4. If {x n } is a sequence in A that converges to x, then x ∈ A.
Proof. Exercise.
Notice that Theorem 3.1 is an if and only if theorem that characterizes the T1
property with a statement about sequences. One can ask whether sequences suffice to characterize Hausdorff spaces, continuous functions, or closed sets. That is,
whether Theorems 3.2, 3.3, and 3.4, have if and only if versions. In general, the
answer is no.
Example 3.4. Sequences don’t suffice to detect Hausdorff spaces. Consider R with
the cocountable topology. This space is not Hausdorff, but convergent sequences
have unique limits.
Example 3.5. Sequences don’t suffice to detect continuous functions. Let X =
[0, 1][0, 1] := {functions f : [0, 1] → [0, 1]} with the product topology and let Y
be the subspace ofRX consisting of integrable functions. The function I : Y → R
1
defined by I ( f ) = 0 f is not a continuous function but {I ( f n )} → I ( f ) whenever
{ fn} → f .
Example 3.6. Sequences don’t suffice to detect closed sets. Let X = [0, 1][0, 1] :=
{functions f : [0, 1] → [0, 1]} with the product topology and let A be the subset of
X consisting of functions whose graphs are “sawtooths” with vertices on the x axis
at a finite number of points {0,r 1 , . . . ,r n , 1} and spikes of height 1 as in the picture
below. The zero function is in A but there is no sequence { f n } in A converging to it.
34
3 Limits of sequences and nets
1
0
r1 r2
r3
r4
1
The spaces in these examples have too many open sets around each point for their
topological properties to be adequately probed by sequences. For spaces without
too many open sets around each point, sequences do suffice to characterize their
properties.
Definition 3.3. Let X be a space. A collection of open sets C is called a neighborhood base for x ∈ X if for every open set O containing x, there exists an open set
U ∈ C with x ∈ U ⊆ O. A space X is called first countable if and only if every
point has a countable neighborhood base. A space X is called second countable if
and only if it has a countable basis.
Example 3.7. An n-dimensional manifold is a second countable Hausdorff topological space with the property that every point has a neighborhood homeomorphic to
Rn .
Example 3.8. Every metric space is first countable since the open balls around x of
radius 1, 12 , 13 , . . . form a countable neighborhood base. The spaces I I and R with
the cocountable topology are not first countable.
In first countable spaces such as metric spaces, sequences do suffice to characterize properties of spaces, subsets, and functions. Here are the three theorems.
Theorem 3.5. Let X be a first countable space. Then X is Hausdorff if and only if
every sequence has at most one limit.
Proof. Suppose that X is first countable. If X is not Hausdorff, there exist points x
and y that cannot be separated by open sets. Let U1 ,U2 , . . . be a neighborhood base
of x and V1 ,V2 , . . . be a neighborhood base for y. For every n there choose a point
x n ∈ Un ∩ Vn , ∅. The sequence {x n } has a subsequence that converges to x and to
y.
Theorem 3.6. Suppose X and Y are first countable and f : X → Y is a function.
The function f : X → Y is continuous if and only if for every sequence {x n } in X
with {x n } → x, the sequence { f (x n )} → f (x).
3.3 Nets and three theorems about them
35
Proof. Exercise.
Theorem 3.7. Let X be a first countable space and let A ⊆ X. A point x ∈ A if and
only if there exists a sequence {x n } in A with {x n } → x.
Proof. Exercise.
3.3 Nets and three theorems about them
The reason sequences characterize closure and continuity in first countable spaces,
but not arbitrary spaces is because sequences are countable. For arbitrary spaces, it’s
better to work with nets.
Definition 3.4. A directed set is a pair (S, ≤) where S is a set and ≤ is a relation on
S satisfying:
• for all s ∈ S, s ≤ s
• for all s,t,u ∈ S, s ≤ t and t ≤ u imply s ≤ u
• for all s,t ∈ S, there exists u ∈ S with s ≤ u and t ≤ u.
So a directed set is a set equipped with a reflexive, transitive, directed relation.
Note that we are not assuming that a directed set be a directed poset—it is not
necessary to assume that the relation be anti-symmetric as is done in [21]. Example
3.11 below involves a directed set that is not anti-symmetric.
Example 3.9. The pair (N, ≤) where ≤ means “less than or equal to” defines a directed set.
Example 3.10. Fix an interval [a, b] in R with a < b. Then (P, ≤) defines a directed
set where P is the set of partitions of [a, b] and ≤ is defined by P ≤ Q iff Q is a
refinement of P. One might say that the partitions of an interval can be directed by
refinement.
Example 3.11. Let X be a metric space and fix a point x ∈ X. Then we can define a
directed set by (X, ≤) where y ≤ z iff d(y, x) ≥ d(z, x). Observe that this relation
is not in general anti-symmetric. One might say that the points of X are directed
toward x.
Example 3.12. Let X be a topological space and let x ∈ X. We define a directed set
(U , ≤) where U is the set of neighborhoods of x and for two neighborhoods U and
U 0, we say U ≤ U 0 iff U 0 ⊆ U. One might say that the neighborhoods of a point
form a directed set under reverse inclusion. This directed set (U , ≤) is a poset hence
a category.
Definition 3.5. A net in a set X is a function x : S → X where S is a directed set.
We usually write x s for x(s) and may denote the net by {x s }. If X is a topological
space and z ∈ X, we say that a net {x s } converges to z and write {x s } → z if and
only if for all open sets U with z ∈ U there exists a t ∈ S so that for all t ≤ s,
x s ∈ U.
36
3 Limits of sequences and nets
Example 3.13. If {x n } is a sequence and as a sequence {x n } → x, then {x n } → x as
a net.
Example 3.14. Here’s an example that was one of the primary motivations for the
definition of a net in [19, 20]. (Convergence of nets is sometimes referred to as
“Moore-Smith” convergence). Let f : [a, b] → R be a bounded function and let P
be the partitions of [a, b] made into a directed set by refinement. We define two nets
UP and L P (for upper and lower) in R as follows: For any partition P = {a = x 0 <
x 1 < . . . < x n = b} define
UP =
n
X
* sup f (x) + (x i − x i−1 ) and L P =
i=1 , x ∈[ x i−1, x i ]
-
n
X
i=1
inf
x ∈[ x i−1, x i ]
!
f (x) (x i − x i−1 ).
Then the nets UP and L P converge to the upper and lower integral of f respectively.
The function f is Riemann integrable if and only if the nets {UP } and {L P } converge
Rb
to the same value, in which case that value is the called integral a f .
Example 3.15. Let X be a topological space and let x ∈ X. For every neighborhood
U of x, choose a point xU ∈ U. Then {xU } defines a net in X and {xU } → x, where
the directed set is the one in Example 3.12.
This last example is the key observation behind the following theorems. We prove
one, and leave the others as exercises.
Theorem 3.8. A space is Hausdorff if and only if limits of convergent nets is unique.
Proof. Exercise.
Theorem 3.9. A function f : X → Y is continuous if and only if for every net {x s }
in X with {x s } → x, the net { f (x s )} → f (x).
Proof. Exercise.
Theorem 3.10. A point x ∈ A if and only if there exists a net {x s } in A with {x s } →
x.
Proof. Suppose that {x s } is a net in A with {x s } → x. Then every open set around
x contains x s for some s and hence contains points of A, proving that x ∈ A.
Conversely, if x ∈ A, every neighborhood U of x contains a point, call it xU , of
A. This defines a net {xU } in A converging to x.
3.4 Tychonoff’s Theorem
The goal of this section is to prove
Tychonoff’s Theorem. Given any collection {Xα }α ∈ A of compact spaces, the prodQ
uct α ∈ A Xα is compact.
3.4 Tychonoff’s Theorem
37
It is easier to prove that the product of finitely many compact spaces is compact
than it is to prove the general case. For example, In Munkres’ Topology [21], compactness is introduced in Chapter 3, where it is proved that the product of finitely
many compact spaces is compact (Theorem 26.7), and the proof is of the general
case for arbitrary products (Theorem 37.3) is postponed until Chapter 5, with a full
chapter on countability and separation interrupting. Schaum’s Outline [15] states
Tychonoff’s theorem in Chapter 12, but the proof is banished to the exercises. One
must use the axiom of choice (or its equivalent) to prove the general case (see 3.13).
It’s worth noting that there’s a two line proof of Tychonoff’s theorem in [28] using
“ultrafilters,” due originally to Cartan [4], but it would be rather more than two lines
to develop that machinery for just this theorem.
3.4.1 Preliminaries from set theory
There’s no way to avoid some set-theoretic details. For convenience, we restate the
axiom of choice.
The Axiom of Choice. Given any collection of nonempty sets {Xα }α ∈ A , the product
Q
α ∈ A X α is nonempty.
We’ll use the interpretation of the product as a set of functions, so the axiom of
choice says that there exists a function f : A → ∪α ∈ A Xα with f (α) ∈ Xα for each
α ∈ A. An equivalent formulation of the axiom of choice is Zorn’s Lemma. We’ll
give a proof that Zorn’s lemma is equivalent to the axiom of choice that involves
“partial” choice functions, which is quite similar to the proof of Tychonoff’s theorem
later. Zorn’s lemma involves some of terminology related to posets, which we now
recall.
Let P be a partially ordered set. A subset C of P is called a chain iff for every
a, b ∈ C either a ≤ b or b ≤ a. An element b ∈ P is called an upper bound for a
subset A ⊂ P provided a ≤ b for all a ∈ A. A subset A ⊆ P has an upper bound
if and only if there exists b ∈ P that is an upper bound for A. We say that m is a
maximal element of a partially ordered set P if and only if there exists no element
a ∈ P with m ≤ a and a , m. One can write a < b if a ≤ b and a , b, and one
can use the notation ≥ and > with the obvious meaning. Then one can say m is a
maximal element if there is no element a with a > m.
Zorn’s Lemma. If every chain in a nonempty partially ordered set P has an upper
bound, then P has a maximal element.
Theorem 3.11. The axiom of choice ⇔ Zorn’s lemma.
Proof. Assume the axiom of choice and let P be a nonempty partially ordered set
in which every chain has an upper bound. For any a ∈ P, define a set Ea := {b ∈
P : a < b}. If Ea = ∅ for any a, then a is a maximal element of P and we’re done.
38
3 Limits of sequences and nets
If, however, Ea , ∅ for any a, then the axiom of choice says there is a function
f : P → ∪a ∈P Ea with f (a) ∈ Ea . This means that f (a) > a for every a. So, we
create a chain
a < f (a) < f ( f (a)) < · · · .
We know this chain has an upper bound since every chain in P has an upper bound.
Call it b. Then f (b) > b and we can extend the chain
a < f (a) < f ( f (a)) < · · · < b < f (b) < f ( f (b)) < · · ·
Now this chain has an upper bound, call it c and we can add c, f (c), f ( f (c)), . . . to
the chain. We can continue transfinitely, which shows that that cardinality of P is
greater than that of any other set, which is impossible. Thus, the axiom of choice
implies Zorn’s lemma.
To prove that Zorn’s lemma implies the axiom of choice, let {Xα }α ∈ A be a
collection of nonempty sets. Define a partial choice function to be a function
f : I → ∪α ∈I Xα where I ⊆ A. The set P of partial choice functions is partially
ordered by extension: for two partial choice function f and g one has f ≤ g iff the
domain of f is a subset of the domain of g and they agree on their common domain.
If C is a chain of partial choice functions, then the union of the functions in C is an
upper bound for C. (The union of a chain of partial choice functions is a function
whose domain is the union of the domains of the functions in C and whose value at
a point is the common value at that point of any one of the functions in C defined at
the point.) Then Zorn’s Lemma implies that there is a maximal element of P, which
must be a choice function with domain A. This function satisfies the conclusion of
the axiom of choice.
There are a few other statements that are equivalent to the axiom of choice. One is
the Hausdorff maximum principle which states that every partially ordered set has a
maximal chain. Here, a maximal chain means a chain that is not properly contained
in any other chain. Another statement equivalent to the axiom of choice is the well
ordering principle which states that every set can be well ordered. A well ordering
is a partial order that is a total order (so every two elements are comparable) for
which every nonempty subset has a least element. The natural numbers N with ≤ is
well ordered. It’s more challenging to try and think of a well ordering on R, which
puts in mind the chapter quote by Jerry Bona
The Axiom of Choice is obviously true, the well-ordering theorem is obviously false; and
who can tell about Zorn’s Lemma?
It’s worth pointing out the inconsistent use of terminology here: equivalent statements might be axioms, lemmas, or even principles.
3.4.2 Nets and compactness
We introduce the notion of a cluster point of a nets.
3.4 Tychonoff’s Theorem
39
Definition 3.6. Let X be a space and let {x α }α ∈Λ be a net. We say that a point x ∈ X
is a cluster point (or accumulation point) of the net {x α }α ∈Λ if and only if for every
open set U containing x and every α ∈ Λ, there exists β ∈ Λ with α ≤ β and
x β ∈ U.
Theorem 3.12. Let X be a topological space. The following are equivalent:
(a) X is compact.
(b) Every collection of closed subsets of X with the FIP has nonempty intersection.
(c) Every net in X has a cluster point.
Proof. (a) ⇒ (b) is Theorem 2.14.
(b) ⇒ (c). Let {x α }α ∈Λ be a net. For each α ∈ Λ define
Fα = {x β : α ≤ β} and Eα = Fα .
Since Λ is a directed set, if α, α 0 ∈ Λ there exists β ∈ Λ with α ≤ β and α 0 ≤ β.
Then x β ∈ Fα ∩ Fα0 ⊆ Eα ∩ Eα0 . It follows that {Eα } is a collection of closed sets
with the FIP. Therefore, there is an element x ∈ ∩Eα . We claim that x is a cluster
point of the net {x α }. To see this, let U be an open set containing x and let α ∈ Λ.
Since x ∈ Eα = Fα , there is a point y ∈ Fα ∩ U. By definition of Fα , y = x β for
some α ≤ β, as needed.
(c) ⇒ (a). Let {Uα }α ∈Λ be a collection of open sets for which no finite subset
covers X. We’ll prove that {Uα } is not a cover of X. Let D be the set of finite subsets
of Λ directed by inclusion: F ≤ G ⇔ G ⊆ F. For each F ∈ D, ∪α ∈F Uα is not
a cover of X, therefore there exists a point x F ∈ X \ (∪α ∈F Uα ). The assignment
F 7→ x F defines a net in X, which by hypothesis has a cluster point, call it x. We
claim that x < ∪α ∈ΛUα . To see this, fix α ∈ Λ. Then the singleton set {α} ∈ D,
therefore, for any open set U containing x, there is a finite set F with {α} ⊆ F and
a point x F ∈ U. This says that x F ∈ X \ (∪ β ∈F Uβ ) ⊆ X \ Uα . This proves that
x ∈ X \ Uα = X \ Uα . Since this is true for every α, x < ∪α ∈ΛUα .
Definition 3.7. A subnet of a net f : S → X is the composition f ◦ φ where φ :
T → S is an increasing cofinal function from a directed set T. Increasing means that
φ(a) ≤ φ(b) whenever a ≤ b and cofinal means that for every s ∈ S, there exists an
t ∈ T with s ≤ φ(t).
Notice that a net has a convergent subnet if and only if it has a cluster point, so
compactness could be characterized by convergent subnets:
Corollary 3.1. A space is compact if and only if every net has a convergent subnet.
The use of nets to characterize compactness brings to mind Theorems 3.2, 3.3,
and 3.4 stated earlier. One might ask whether sequences suffice to characterize compactness in first countable spaces. The answer is no, not in general. There are first
countable spaces that are not compact which are sequentially compact meaning that
every sequence has a convergent subsequence. The reader who is interested in looking for such a space will want to know that metric spaces are compact if and only if
they are sequentially compact.
40
3 Limits of sequences and nets
3.4.3 A proof of Tychonoff’s Theorem
Here, we give the proof due to Chernoff in 1992 [5].
Proof (Proof of Tychonoff’s theorem). Let {Xα }α ∈ A be a family of compact spaces,
Q
let X = α ∈ A Xα and let { f d }d ∈D be a net in X. As in the proof of Theorem 3.4.1,
Q
we define an element of α ∈I Xα where I ⊆ A to be a partial function. We say that
a partial function f is a partial cluster point if it is a cluster point of the net { f d | I }
for some I ⊆ A. If there is a partial cluster point with domain I = A, then we have
found a cluster point of the net { f d } and proved that X is compact.
Let P be the set of partial cluster points ordered by inclusion. Note that P , ∅
for if we let α ∈ A be one index and set I = {α}, then { f d (α)} is a net in Xα . Since
Xα is compact, there is a cluster point p ∈ Xα of the net { f d (α)}. Then for I = {α},
the partial function f : I → X defined by f (α) = p is a partial cluster point. Also,
every chain in P has an upper bound since the union of the partial cluster points in
a chain will also be a partial cluster point. Thus P satisfies the hypotheses of Zorn’s
Lemma.
Let g be a maximal element of P. If the domain I of g is equal to A then we
are done. If the domain of g , A, choose an index α ∈ A \ I. Since g is a cluster
point of the net { f d | I }d ∈D , there’s a subnet { f φ(e) | I }e ∈E converging to g. Since Xα
is compact, there’s a cluster point p ∈ Xα of the net { f φ(e) (α)}e ∈E . Then h defined
by

p
if β = α
h( β) = 
 g( β) if β ∈ I

is a partial cluster point with domain I ∪ {α} extending g. This contradicts the maximality of g. Therefore, the domain of g is all of A and the proof is complete.
3.4.4 Tychonoff’s theorem implies the axiom of choice
We used Zorn’s lemma to prove the Tychonoff theorem. In 1950, Kelley proved
that the Tychonoff theorem is equivalent to the axiom of choice [13]. In order to
prove that the Tychonoff theorem implies the axiom of choice, one begins with
an arbitrary collection of sets and then creates a collection of compact topological
spaces. Then, the compactness of the product leads to the existence of a choice
function. Originally, Kelley used an augmented cofinite topology. Here is an easier
proof.
Theorem 3.13. The Tychonoff theorem ⇔ the axiom of choice.
Proof. We used Zorn’s lemma to prove Tychonoff’s theorem, which shows that Tychonoff’s theorem is implied by the axiom of choice.
To prove that Tychonoff’s theorem implies the axiom of choice, let {Xα }α ∈ A be a
collection of nonempty sets. We need to make a bunch of compact spaces so we can
3.4 Tychonoff’s Theorem
41
apply the Tychonoff theorem. First, let Yα = Xα ∪ {∞α }; we add a new element to
Xα called “∞α ” We make each Yα into a topological space by defining the topology
to be {∅, {∞α }, Xα ,Yα }. Note that Yα is compact—there are only finitely many open
Q
sets so every open cover is finite. So, by Tychonoff’s theorem, Y := α ∈ A Yα is
compact.
Now consider a collection of open sets {Uβ } β ∈ A of Y where Uβ is the basic open
set in Y obtained by taking the product of all Yα ’s for α , β and in the β factor
put the open set {∞ β }. Notice that any finite subcollection {Uβ1 , . . . ,Uβ n } cannot
n U : choose a partial
cover Y for the function f defined as follows is not in ∪i=
1 βi
Qn
function f ∈ i=1 X β i which is possible without the axiom of choice since the
product is finite. Then extend f to a function f ∈ Y be setting f (α) = ∞α for all
α , β1 , . . . , β n , which is possible since we’re not making any choices.
Therefore, the collection {Uβ } cannot cover Y . So, there is a function f ∈ Y not
in the ∪α ∈ AUα . This says that for no α ∈ A does f (α) = ∞α . Therefore, f (α) ∈ Xα
for each α, which is a desired choice function.
Exercises
3.1. Prove that R with the cocountable topology (sets with countable complement
are open) is a non-Hausdorff space in which convergent sequences have unique limits.
3.2. Show that the directed sets defined in Example 3.12 above are in fact directed
sets.
3.3. Check all the details of Example 3.5
3.4. Check all the details of Example 3.6
3.5. Prove Theorem 3.8.
3.6. Prove Theorem 3.9.
3.7. Here are two variations of Hausdorff. Call a space KC all its compact sets are
closed. Call a space U S if the limits of convergent sequences are unique. Prove that
Hausdorff implies KC implies U S, but that the implications are strict [27].
3.8. Show that a countable intersection of open dense sets in a complete metric space
is dense. (This is called the Baire category theorem.)
3.9. Let X be a compact space and let { f n } be an increasing sequence in hom(X, R).
Prove that if { f n } converges pointwise then { f n } converges uniformly.
3.10. A sequence is an example of a net. Show that a subsequence of a sequence
is a subnet, but not all subnets of a sequence are subsequences. For an interesting
example, use the family of sawtooth functions from Example 3.6 whose corners
have rational coordinates.
42
3 Limits of sequences and nets
3.11. Let X be a topological space. If every infinite subset of X has a limit point, X is
said to be limit point compact. If every sequence in X has a convergent subsequence,
X is said to be sequentially compact.
(a)Prove that compactness implies limit point compactness in any space X. Give an
example to show the implication is strict.
(b)Prove that limit point compactness implies sequential compactness if X is first
countable and T1 .
Chapter 4
Categorical limits and colimits
A comathematician is a device for turning cotheorems into ffee.
—
Subspaces, quotients, products, and coproducts are particular examples of categorical limits and colimits. But there are many other important constructions
(pushouts, pullbacks, direct limits, ...) that are categorical limits and colimits, so
it’s valuable to discuss the general notion.
4.1 Diagrams are functors
First, realize that diagrams in a category are functors. For example, a diagram like
this
f
→Y
X
→
←
←
g
→
←
h
Z
in a category C consists of a choice of three objects X, Y , and Z and some morphisms
f : X → Y , g : X → Z and h : Z → Y , and we know, since the diagram commutes,
that h f = g. A picture like this
→ •
→
←
←
•
←
→
•
43
44
4 Categorical limits and colimits
is a small category D containing three objects pictured as bullets and three morphisms pictured as arrows. The three identity morphisms are not pictured, but are
in the category. Composition is determined by setting the composition of the two
diagonal arrows to be the horizontal arrow. A functor F : D → C involves a choice
of three objects and three morphisms and must respect composition. Conclusion:
is a diagram
g
←
→
←
•
h
Z
-
,
→Y
→
→
f
←
←
←
X
+/
/
−→ C///
//
←
→ •
→
*. •
.
a functor ...
..
Definition 4.1. Let D be a small category. A D-shaped diagram in a category C
is a functor D → C. A morphism from a D-shaped diagram to another D-shaped
diagram is a natural transformation of functors. If the categories C and D are understood, we might just say diagram instead of D-shaped diagram in C.
One can view an object A of C as a D-shaped diagram by considering it as the
constant functor D → C that sends every object in D to A and every morphism
to id A . Therefore, one can talk about maps from an object to a diagram or from a
diagram to an object. Sometimes maps from an object to a diagram are called cones.
Unwinding the definition, a map from an object A to a diagram F : D → C consists
of a collection of morphisms from A to every object in the diagram
{ f X : A → F (X ) : X is an object of D}
that respect composition of morphism in the diagram:
F ( f ) f X = f Y whenever f : X → Y is an arrow in D.
For example, a map from an object A to the diagram above consists morphisms
f X : A → F X, f Y : A → FY , f Z : A → F Z that fit together as pictured below.
f
←
A
←
←
g
→ FY
→
fY
fX
fZ
←
→
→
FZ
→
←
→
←
FX
h
4.3 Examples
45
4.2 Limits and colimits
Now, we can define the limit of a D-shaped diagram F : D → C.
Definition 4.2. A limit of the diagram F : D → C is a map f from an object A to the
diagram satisfying the universal property that for any other map g from an object B
to the diagram, there exists a unique morphism h : B → A such that f X h = gX for
all objects X in D.
→ FY
→
fY
fX
←
A
→
gX
→
→
Ff
←
→
←
FX
gY
h
←
←
←
B
Similarly, one can talk about maps from a diagram F : D → C to an object A in
C. Then, one has
Definition 4.3. A colimit of the diagram F : D → C is a map f from the diagram
to an object A satisfying the universal property that for any other map g from the
diagram to an object B, there exists a unique morphism h : A → B such that
h f X = gX for all objects X in D.
→
A
←
gX
→ FY
←
fY
fX
←
←
Ff
→
←
←
FX
gY
h
→
→
→
B
4.3 Examples
To help understand limits and colimits, we work through some commonly occuring
examples.
46
4 Categorical limits and colimits
4.3.1 Initial and terminal objects
The limit of the empty diagram is a terminal object and the colimit of an empty
diagram is an initial object. If D has no non-identity morphisms, then a diagram
D → C is just a collection of objects parametrized by D. In this case, the limit of the
diagram is the product of the objects and the colimit of the diagram is the coproduct
of the objects. The reader is invited to check that the universal properties of the
product and coproduct as given in Chapter 0 show that they are the limit and colimit
of discrete diagrams.
Many times when one is interested in the limit of a diagram, the colimit of the
diagram will be trivial, or vice-versa.
For example, the colimit of a diagram that has a terminal object Y is just the
object Y , together with the morphisms in the diagram. For example, the object Y is
terminal in the diagram
f
→Y
→
←
←
X
g
→
←
h
Z
and the colimit of this diagram is just Y with the map of the diagram given by
f : X → Y , h : Z → Y and idY : Y → Y . It has the universal property since
for any object S, a map from the diagram to S includes a map from Y to S making
everything commute—that map is the one satisfied by the universal property of Y
being the colimit.
→
g
←
h
←
→
Z
→ S
→
→
←
←
→Y
←
←
f
→
←
X
Similarly, the limit of a diagram with an initial object X is just the initial object
X, together with the morphisms in the diagram.
4.3.2 Pushouts
The colimit of the diagram
4.3 Examples
47
Z
→ X
←
←
g
→
f
Y
is called the pushout of X and Y along the morphisms f and g. For an example, for
any two sets A and B, the intersection A ∩ B is a subset of A and B. We have the
diagram
A ∩ B -← → A
-←→
B
and the pushout is the union A ∪ B. This means that the union fits into the diagram,
and for any other set S that fits into the diagram, there’s a unique function from
A ∪ B → S as pictured by the dashed line below:
→ A
→
←
-←→
→ A∪ B
→
←
←
←
B
←
A ∩ B -←
→
→ S
The universal property of the pushout in this example says that a function f : A ∪
B → S is the same as a functions A → S and B → S that agree on A ∩ B.
In Top, pushouts describe the constructions of gluing spaces together along maps.
Set theoretically, the pushout of maps f : Z → X and g : Z → Y is X t Z Y =
`
X Y/∼ where f (z) ∼ g(z). The pushout becomes a topological space as a quotient
space of the coproduct. This pushout satisfies the universal property described by
this diagram:
→ X
←
←
←
g
←
←
→ X tZ Y
→
→
←
iY
→
iX
→
Y
f
←
Z
→W
The pushout topology is (first characterization) the coarsest topology for which the
maps i x : X → X t Z Y and iY : Y → X t Z Y are continuous. Alternatively (second
characterization) the pushout topology is determined by specifying that maps from
the pushout to any space W are continuous if and only if the maps X → W and
Y → W obtained by precomposing with i X and iY are continuous. The other way
around: a map from the pushout to a space W are specified by maps a : X → W and
b : Y → W with f a = gb.
48
4 Categorical limits and colimits
Pushout diagrams like this are commonly used to describe the space obtained by
attaching a disc D n to a space X along a map f : S n−1 → X.
f
→ X
←
←
←
S n−1
iX
←
Dn
iY
→
→
i
→ X tS n−1 D n
In this case, the map S n−1 ,→ D n is often understood as the inclusion, and one
describes the pushout succintly by saying “the disc D n is attached to X via the
attaching map f ” and writes X t f D n .
4.3.3 Pullbacks
The limit of the diagram
←
X
←
Y
→
f
g
→ Z
is called the pullback of X and Y along the morphisms f and g. In Set and Top, the
pullback is realized by X × Z Y = {(x, y) : f (x) = g(y)}.
You might encounter a statement like “the map p : Y → X is the pullback of
π : E → B by the map f : X → B.” This means that p fits into a square with f and
π as pictured
→ E
Y
←
←
←
π
←
X
f
→
→
p
→ B
and Y together with p : Y → X and the unnamed map is the pullback of the rest of
the diagram. The unnamed map Y → E exists and is part of the pullback, but might
not be explicitly mentioned.
4.3.4 Equalizers and coequalizers
The limit of the diagram
←
←
X
f
→Y
g→
4.3 Examples
49
is called the equalizer of f and g. In Set and Top, the equalizer can be realized as
the set E = eq( f , g) = {x ∈ X : f (x) = g(x)} with the inclusion E → X. It’s the
largest subset of the domain X on which the two maps agree. Here’s the picture of
its universal property:
→
←
f
→Y
g→
←
←
→ X
→
←
←
E
S
In algebraic categories, such as Group, Vect, RMod, the equalizer of f : G → H
and the unique map from the initial object 0 : G → H is the called the kernel of f .
The colimit of the diagram
←
←
X
f
→Y
g→
is called the coequalizer of f and g. In Set and Top, the coequalizer is realized as
the quotient of the set Y obtained by identifying f (x) and g(x) for each x ∈ X. It’s
the quotient of the codomain Y by the smallest relation that makes the maps agree.
In algebraic categories, such as Group, Vect, RMod, the coequalizer of f : G → H
and the unique map from the initial object 0 : G → H is the called the cokernel of
f.
4.3.5 Direct and inverse limits
The colimit of the diagram X1 → X2 → X3 → · · · is sometimes called the directed
limit of the {X i } and denoted by lim→ X i , but we prefer to call it the colimit of the
diagram. The colimit is an object X together with maps f i : X i → X that assemble
to be a map from the diagram. In a concrete category in which the objects are sets
with some structure, and the maps X i → X i+1 are injections, one can think of the
diagram as an increasing sequence of objects. You can think of the colimit of the
diagram, if it exists, as the union of the objects.
Example 4.1. In linear algebra, the colimit of N copies of R is the set of sequences
of real numbers for which all but finitely many are zero and is denoted ⊕n ∈N R. This
`
is not the same as the colimit of N copies of R in Top, which is n ∈N R. To make
the vector space ⊕n ∈N R into a topological space, you need to view it in a different
way.
Specifically, X = ⊕n ∈N R is the colimit of the diagram of (vector and topological)
spaces
R → R2 → R3 · · ·
where the map Rn → Rn+1 is given by (x 1 , . . . , x n ) 7→ (x 1 , . . . , x n , 0). Think of the
diagram as an increasing union: R sits inside R2 as the x-axis, then R2 sits inside
50
4 Categorical limits and colimits
R3 as the xy-plane, etc... The colimit X of the diagram is an infinite dimensional
space in which all these finite dimensional spaces sit inside, and is the smallest such
space, meaning that if Y is any other space that has maps X i → Y , these maps factor
through a map X → Y . The space X is realized as the set of sequences (x 1 , x 2 , . . .)
for which all but finitely many x i are nonzero, together with the maps Rn → X
defined by
(x 1 , . . . , x n ) 7→ (x 1 , . . . , x n , 0, 0, . . .)
which identify the Rn ’s with increasing subsets of X. The vector space structure is
addition and scalar multiplication of sequences. The topology on X coming from
the colimit can be described explicitly by saying that a set U of sequences is open if
and only if the intersection U ∩ Rn is open for all n ∈ N. This is the finest topology
that makes the inclusions R n ,→ X continuous.
The limit of the diagram X1 ← X2 ← X3 ← · · · is called the inverse limit of
the {X i } and is denoted by lim← X i . Think of it as the smallest object that projects
down to the X i ’s.
For example, the limit X of the diagram of spaces
R ← R2 ← R 3 · · ·
where the map Rn+1 → Rn is given by (x 1 , . . . , x n , x n+1 ) 7→ (x 1 , . . . , x n ) is the
Q
product n ∈N R, the set of all sequences (x 1 , x 2 , x 3 , . . .) with the product topology.
The projections X → Rn defined by (x 1 , x 2 , . . .) 7→ (x 1 , . . . , x n ) define the map
from X to the diagram, and the topology on X is the coarsest topology making the
maps from X to the diagram continuous. Here, the limit X of the diagram R ←
R2 ← R3 · · · agrees in both Top and Vect.
Notice that the limit of the diagram X1 → X2 → X3 → · · · or the colimit of the
diagram X1 ← X2 ← X3 ← · · · are both just the object X1 .
4.4 Completeness and cocompleteness
A category is called complete if it contains the limits of diagrams and is called
cocomplete if it contains the colimits of diagrams.
The categories Set and Top are complete and cocomplete. In Set, one can construct the colimit of any diagram by taking the disjoint union of every set in the
diagram and then quotienting by the relations required for the diagram to map into
the resulting the set. In Top, this set gets the quotient topology of the disjoint union.
This topology is the finest topology for which all the maps involved in the map from
the diagram are continuous.
To construct the limit of any diagram of sets, first take the product of all the sets
that appear in the diagram. The product then maps to all objects in the diagram.
The limit of the diagram is simply the subset of the product so that the projection
maps to the objects assemble to be a map to the diagram. In Top, this set gets the
4.4 Completeness and cocompleteness
51
subspace topology of the product. This topology is the coarsest topology for which
all the maps involved in the map to the diagram continuous.
Seeing how to define an arbitrary colimit of sets as a quotient of the disjoint
union, or how to define an arbitrary limit as a subspace of the product, gives the idea
of how to prove the following theorem:
Theorem 4.1. If a category has products and equalizers then it is complete. If it has
coproducts and coequalizers then it is cocomplete.
Proof. Here’s how to construct the colimit of a diagram in a category with coproducts and coequalizers. Proceed in two steps. First, take the coproduct Y of all the
objects Xα in the diagram that have morphisms Xα → X β from them (there may
multiple copies of Xα ’s) and take the coproduct Z of all the objects X β that appear
in the diagram (just one copy each):
a
a
Y :=
Xα
Z :=
Xβ
β
X α →X β
There are two maps Y → Z. One, call it f , is defined by taking the coproduct of the
morphisms in the diagram and one defined simply by identities. The coequalizer of
these two maps
←
←
Y
f
→ Z
→
id
is the coproduct of the diagram.
The idea for limits is similar.
We end this chapter with a definition.
Definition 4.4. A functor is continuous if it takes limits to limits; it is cocontinuous
if it takes colimits to colimits.
Exercises
4.1. Let f : Y → X be an embedding of a space Y into a space X. Construct a
diagram for which Y (and the map f ) is a limit. Hint: Exercise 1.12 shows that
quotients are coequalizers, hence colimits.
4.2. Define the infinite dimensional sphere S ∞ to be the colimit of the diagram
S 0 ,→ S 1 ,→ S 2 ,→ S 3 ,→ · · ·
Prove that S ∞ is contractible.
4.3. [16, p 72, #9] From a diagram
52
4 Categorical limits and colimits
f
→Y
←
←
←
X
g→
h
→ Z
construct a commutative square
( f , g)
X
→Y
←
←
←
`
→
→
X
→ Z
h
←
X
Prove that the first diagram is a coequalizer precisely if the second is a pushout.
From a commutative square
X
←
←
→Y
←
f
←
X
→
p
→
g
q→
Z
construct a diagram
f
→ B `C
g→
(p,q)
←
←
←
X
→ Z
Prove that the first diagram is a pushout if and only if the second is a coequalizer.
Conclude that a category that has pushouts and coproducts is coclosed. Give a similar argument to prove that a category that has pullbacks and products is closed.
4.4. [16, p 72, #4] In any category, prove that f : X → Y is an epimorphism if and
only if the following square is a pushout:
→Y
←
f
←
←
X
f
←
Y
→
→
idY
→Y
idY
Chapter 5
Adjunctions and the compact open topology
Birds fly high in the air and survey broad vistas of mathematics out to the far horizon.
They delight in concepts that unify our thinking and bring together diverse problems from
different parts of the landscape. Frogs live in the mud below and see only the flowers that
grow nearby. They delight in the details of particular objects, and they solve problems one
at a time.
— Freeman Dyson
In this chapter we introduce adjoint functors and use them to highlight several
constructions. Quite a few pages are devoted to putting an appropriate topologies on
function spaces and some of the difficulties involved. In this chapter, we’ll be both
birds and frogs. Often times, a categorical point of view highlights and elevates the
important properties that characterize an object or construction, but fails to establish
that such objects exist. Existence often requires getting down in the mud.
5.1 Adjunctions
We begin with the definition and discuss a few examples.
Definition 5.1. Let C and D be categories. An adjunction between C and D is a pair
of functors L : C → D and R : D → C together with an isomorphism
homD (LX,Y ) −→ homC (X, RY )
(5.1)
for each object X in C and each object Y in D that is natural in both components.
The functor L is called the left adjoint and the functor R is called the right adjoint.
The adjunction is often denoted succinctly by
L: C D : R.
To say that the isomorphism in (5.1) is “natural” in both components means that
it arises via natural transformations. More precisely, for each object X in C there are
53
54
5 Adjunctions and the compact open topology
two functors D → Set
homD (LX, −) and homC (X, R−)
and for each object Y in D there are two functors Co p → Set
homD (L−,Y ) and homC (−, RY ).
Then saying the isomorphism homC (LX,Y ) −→ homD (X, RY ) is “natural in both
coordinates” means that there are natural transformations of functors
homD (LX, −) −→ homC (X, R−) and homD (L−,Y ) −→ homC (−, RY ).
Z
Example 5.1. For any sets X,Y , and Z, the bijection Y X ×Z −→ Y X arises from
an adjunction. The functor X × − : Set → Set is a left adjoint of the functor
hom(X, −) : Set → Set. To see it clearly, fix a set X and define two functors
L = X × − : Set → Set and R = hom(X, −) : Set → Set.
Then
Z
homSet (L Z,Y ) = Y X ×Z Y X = homSet (Z, RY )
The setup L: Set Set : R is an adjunction.
5.1.1 The unit and counit of an adjunction
Suppose L: C D : R is an adjunction with adjunction isomorphism
φ X,Y : homD (LX,Y ) −→ homC (X, RY )
(5.2)
By setting Y = LX, we have an isomorphism
φ X, L X : homD (LX, LX ) −→ homC (X, RLX ).
Via the adjunction isomorphism, the morphism id L X in the category D corresponds
to a morphism X → RLX in the category C, which defines a natural transformation
η : idC → RL
called the unit of the adjunction.
Similarly, for X = RY , under the isomorphism
φ RY,Y : homD (LRY,Y ) −→ homC (RY, RY )
5.2 Free-Forgetful adjunction in algebra
55
the morphism id RY in C corresponds to a morphism LRY → Y which defines a
natural transformation
: LR → idD
called the counit of the adjunction. Understanding the counit and unit of an adjunction helps to understand the isomorphisms (5.2) and the universal properties. For
example, suppose X ∈ C. For any Y ∈ D and f : X → RY , there exists a unique
fˆ : RLX → Y so that fˆη = f . Here’s the picture
→
RLX
←
X
←
←
η
fˆ
f
→
→ RY
The map fˆ is obtained by applying R to the composition
Ff
→ LRY
Y
←
←
LX
→ Y.
Example 5.2. Let’s look at the unit and counit of the adjunction L : Set Set : R
in Set where
L = X × − : Set → Set and R = hom(X, −) : Set → Set.
The unit of this adjunction is the evaluation map X × Y X → Y defined by
ev(x, f ) = f (x). The counit is the map Z → (X × Z ) X defined by z 7→ (−, z),
the map X → X × Z defined by x 7→ (x, z).
5.2 Free-Forgetful adjunction in algebra
Often, free constructions in algebra (free modules, free groups, free abelian groups,
free monoids, etc...) are defined by universal properties. To be definite, we’ll just
consider the free group but the modifications for other free constructions are easy.
Here’s the way a free group is commonly defined.
A free group on a set S is a group F S together with a map of sets η : S → F S satisfying
the property that for any group G and any map of sets f : S → G there exists a unique
group homomorphism fˆ : F S → G so that fˆη = f .
The picture can help
→
FS
←
S
←
←
η
fˆ
f
→
→ G
but the picture is also confusing. After all, some objects in the picture are sets,
some are groups, some arrows are set maps, some are group homomorphisms. One
56
5 Adjunctions and the compact open topology
gets quite a bit of of clarification by observing that there is a forgetful functor U :
Group → Set that assignes to any group the set underlying the group and to any
group homomorphism to itself, viewed as a set map. Then one may define a free
group on a set S to be a group F S and a map η : S → U F S with the property that
for all groups G and maps f : S → UG there exists a unique map fˆ : F (S) → G so
that f = U fˆη. The right picture is in Set:
S
←
←
η
U fˆ
←
→
UFS
f
→
→ UG
Notice that the “there exists” part of the definition of a free group says that for every
group G, the map hom(S,UG) → hom(F S, G) is surjective. The “unique” part of
the definition says that hom(S,UG) → hom(F S, G) is injective. The fact is that
“free” and “forgetful” form an adjoint pair F : Set Group : U providing the
isomorphism
hom(S,UG) ' hom(F S, G)
and the unit of which defines the inclusion η : S → U F S.
Remark 5.1. The universal property defining a free group can also be understood
within the context of set maps S → UG for all groups G. So, one could make a
category out of this context. Let’s call this category U S . An object in U S is a group
f
G and a set map f : S → UG. A morphism between two objects S → UG and
f0
S → UG 0 is a group homomorphism φ : G → G 0 so that Uφ f = f 0 . That is
S
f0
f
UG
Uφ
UG 0
Then,
A free group on a set S is an initial object in the category U S .
The context for the universal property is put into a category U S that is built out of
the undisguised material involved: the set S and the functor U : Group → Set. Then
the universal object is a familiar notion (an initial object) from category theory.
5.3 Compactifications
57
5.3 Compactifications
Definition 5.2. A compactification of a topological space is an embedding of the
space as a dense subspace of a compact Hausdorff space.
So, a compactification of X is a compact Hausdorff space Y and a continuous injection i : X → Y with X i(X ) ⊂ Y and X = Y . Note that only Hausdorff spaces
have compactifications since every subspace of a Hausdorff space is Hausdorff.
Example 5.3. The maps (0, 1) ,→ S 1 defined by t 7→ (cos(2πt), sin(2πt)) and
(0, 1) ,→ [0, 1] are compactifications. For the space X = (0, 1) with the discrete
topology, he map X ,→ [0, 1] is not an embedding, hence not a compactification.
5.3.1 The one-point compactification
A space X has one-point compactification, sometimes called the “Alexandroff” onepoint compactification if and only if X is Hausdorff and locally compact. If a space
has a one point compactification, it’s unique.
To see this, suppose X ,→ Y is a compactification and Y \ X = {p}. The open
neighborhoods of p are precisely the complements of compact subsets of X: the
complement of an open set containing p is a closed subset of a compact space and
so is compact and conversely if K is a compact subset of Y , it is closed, so it’s
complement is an open set containing p. Then, because points of X can be seperated
by open sets from the point p ∈ Y \ X, there’s a neighborhood of every point of X
contained in a compact set, so X is locally compact.
Conversely, beginning with any space X, one constructs a new space by adding
a point p and defining the open neighborhoods of p to be complements of compact
sets in X. If X is locally compact and Hausdorff, the result is a topology on X ∪ {p}
that is compact and Hausdorff having X as a dense subset.
5.3.2 The Stone-Čech compactification
There’s another compactification with good categorical properties called the StoneČech compactification. Let CH be the category with objects consisting of compact
Hausdorff spaces and morphisms being continuous functions. There is a functor
U : CH → Top, which is just the inclusion of compact Hausdorff spaces as a
subcategory of topological spaces; the functor U is the identity on objects and morphisms. There is a functor β : Top → CH that is a left adjoint to U called the
“Stone-Čech compactification.” The construction is outlined as Construction 6.11
in [17] and, in more detail, in Section 38 of [21]. Here, let’s just unwind this functorial description and see what it means. To say that β is a left adjoint of U means
58
5 Adjunctions and the compact open topology
that for every topological space X and every compact Hausdorff space Y , we have a
bijection
hom( β(X ),Y ) hom(X,UY ) = hom(X,Y ).
This says continuous functions f : X → Y from a space X to a compact Hausdorff
space Y correspond precisely to continuous functions fˆ : β(X ) → Y . Specifying the
continuous functions from β(X ) determines the space β(X ) if it exists, but doesn’t
prove it exists. For that, you need a construciton as in the references mentioned
above.
The unit of the Stone-Čech compactification adjunction
β: CH Top :U
defines a morphism η : X → U βX. Since U βX = βX, the Stone-Čech compactification as a left adjoint of U doesn’t just produce a compact Hausdorff space β(X )
from any topological space X, it also produces a continuous function η : X → βX
involved in a universal property: for every map f : X → Y between X and a compact Hausdorff space Y , there is a unique map fˆ : βX → Y so that fˆη = f .
→
βX
←
X
←
←
η
fˆ
f
→
→Y
In the case X is locally compact and Hausdorff, the map e : X → βX is a compactification of X; that is, e : X → β(X ) is an embedding and X = β(X ). Then for any
compact Hausdorff space Y , the map fˆ : βX → Y is the extension of of the map
f : X → Y.
The one point compactification, call it X ∗ , of a locally compact Hausdorff space
X, is convenient to define but it doesn’t have good properties with respect to morphisms. It definitely doesn’t satisfy the condition that
hom(X ∗ ,Y ) hom(X,Y ).
For a simple example, let X = (0, 1) and the one point compactification i : (0, 1) →
S 1 . Let Y = [0, 1] and consider the inclusion f : (0, 1) → [0, 1] which cannot be
extended to a continuous function from S 1 → [0, 1].
→
S1
←
i
←
(0, 1)
f
→ [0, 1]
5.4 The forgetful functor U : Top → Set and its adjoints
59
5.4 The forgetful functor U : Top → Set and its adjoints
The forgetful functor that assigns to any topological space (X τX ) the set X and to
any continuous function f : (X, τX ) → (Y, τY ) the function f : X → Y is both a
left and right adjoint in Top. Define a functor D : Set → Top that assigns to any set
X the spaces (X, τdi scr et e ) with the discrete topology. To any function f : X → Y ,
let D f = f , which is a continuous function. The setup D : Set Top : U is an
adjunction: For any set X and any space Y we have
homTop (DX,Y ) homSet (X,UY )
On the right, we have arbitrary functions from the set X into the space Y , viewed
as a set. On the right, we take continuous functions from DX → Y which are all
functions from X → Y since every function from a discrete space with is continuous.
On the other hand, define a functor I : Set → Top that assigns to a set the same
set with the indiscrete topology and to any function f : X → Y , the same function,
which will be continuous. Then U : Top Set : I is an adjunction: For any space
X and any set Y we have
homSet (U X,Y ) homTop (X, IY )
On the left we have arbitrary functions from X, viewed as a set, to the set Y . On the
right, we have continuous functions X → IY , which are all functions X → Y since
every function into an indiscrete space is continuous.
The universal properties arising from these adjunctions don’t seem very interesting, but the fact that U is both a left and a right adjoint has important consequences.
In particular,
Theorem 5.1. If L : C → D has a right adjoint, then L is cocontinuous. If R : D →
C has a left adjoint, then R is continuous.
Proof. Suppose R has a left adjoint, call it L. We’ll prove R preserves products.
Consider two objects X and Y in D and their product X × Y . We want to prove that
R(X × Y ) is the product of RX and RY in the category C. We have
homC (Z, R(X × Y )) homD (L Z, X × Y )
homD (L Z, X ) × homD (L Z,Y )
homD (Z, RX ) × homD (Z, RY ).
So maps from any object Z in C to R(X × Y ) correspond to pairs maps Z → RX
and Z → RY as needed.
This explains why the constructions of products and coproducts, subspaces and quotients, equalizers and coequalizers, pullbacks and pushforwards in Top must have,
as an underlying set, the corresponding construction in Set: if the construction exists
in Top, the forgetful functor U : Top → Set preserves it.
60
5 Adjunctions and the compact open topology
5.5 The exponential topology
Consider the general problem of equipping the set of continuous functions hom(X,Y )
from a space X to a space Y with a topology. What properties should this topology
have? We take as motivation the following desiradata:
1. For a fixed space X, the functors F : X × − : Top → Top and F = hom(X, −) :
Top → Top should form an adjoint pair. That is, for all spaces Z, we should
have an isomorphism of sets hom(X × Z,Y ) −
→ hom(Z, hom(X,Y )).
2. The adjunction X × −: Top Top :hom(X, −) should give rise to homeomor
phisms of spaces hom(X × Z,Y ) −
→ hom(Z, hom(X,Y )).
3. Composition of continuous functions to define a continuous function of spaces
hom(X,Y ) × hom(Y, Z ) → hom(X, Z ).
Let’s look closer at the first desirable property. Let g : X × Z → Y be continuous. Denote the transpose by g t : Z → hom(X,Y ). In order for g t to be continuous, the topology on hom(X,Y ) should be rather coarse. However, if the topology on hom(X,Y ) is too coarse (think of the indiscrete topology) then the set
hom(Z, hom(X,Y )) will contain too many continuous functions—it will contain
functions that are not the transpose of any continuous map g : X × Z → Y . One
can show that if there exists a topology on hom(X,Y ) so that for any space Z, the
correspondence g 7→ g t defines a bijection of sets
hom(X × Z,Y ) −
→ hom(Z, hom(X,Y ))
then that topology is unique [2]. Let’s call it the exponential topology and let Y X
denote the set hom(X,Y ) with this exponential topology. If X is locally compact and
Hausdorff, this topology coincides with what is called the compact-open topology
[6], which we discuss in some detail in the next section. In addition to X being
locally compact and Hausdorff, if the space Z is Hausdorff, then the second and
third desirable properties hold. To explain how to topologize hom(X,Y ) in more
general settings requires a shift in perspective that we discuss later in Section 5.7.
5.5.1 The compact-open topology
Before we continue, let’s define the compact-open topology and try to give you a
feel for it (see also pages 529-532 of [9]).
Definition 5.3. Let X and Y be topological spaces. For each compact set K ⊂ X
and each open set U ⊂ Y , define S(K,U) = { f ∈ hom(X,Y ) : f (K ) ⊂ U }. The
sets S(K,U) form a sub-basis for a topology on hom(X,Y ) called the compact-open
topology.
For the record, the product topology is usually not an appropriate topology for
hom(X,Y ) since it treats the space X only as an index set—it doesn’t use the topol-
5.5 The exponential topology
61
ogy of X except to identify the continuous functions within the set of all functions X → Y . Still, a sub-basis for the product topology on Y X consists of sets
S(F,U) = { f : X → Y : f (F) ⊂ U } where F ⊂ X is finite and U ⊂ Y is open. That
is, the product topology could is the “finite-open” topology. In the case X has the
discrete topology, all functions X → Y are continuous and the compact-open topology and the product topologies on hom(X,Y ) coincide. More generally, every finite
set is compact, so the compact-open topology is finer than the product-topology. As
a consequence, fewer sequences converge in the compact-open topology than the
product topology.
A closer look at convergence can give you a good feel for the difference between the product topology and the compact-open topology. A sequence of func
tions f n : [0, 1] → [0, 1] converges to a limiting function f in the product topology
if and only if { f n } → f pointwise. A sequence { f n } in hom([0, 1], [0, 1]) converges
to f in the compact-open if and only if { f n } → f uniformly. To see this, consider a
more general situation.
Suppose that X is compact and Y is a metric space. Then hom(X,Y ) becomes a
metric space with the metric defined by
d( f , g) = sup d( f (x), g(x)).
x ∈X
Two functions f , g ∈ hom(X,Y ) are close in this metric if their values f (x) and
g(x) are close for all points x ∈ X. A sequence { f n } in hom(X,Y ) converges to f
in this metric topoplogy if and only if for all > 0 there exists an n ∈ N so that for
all k > n and for all x ∈ X, d( f k (x), gk (x)) < .
Lemma 5.1. Let X be a metric space and let U be open. For every compact set
K ⊂ U, there is an > 0 so that for any x ∈ K and any y ∈ X \ U, d(x, y) > Proof. This is a straightforward argument using the definition of compactness.
Theorem 5.2. Let X be compact and Y be a metric space. The compact-open topology on hom(X,Y ) is the same as the metric topology.
Proof. Let f ∈ hom(X,Y ), > 0, and consider B( f , ). We’ll find an set O, open
in the compact open topology, with f ∈ O ⊂ B( f , ). This will show that B( f , )
is open in the compact open topology, and prove that the compact open topology is
finer
( than the )metric topology. Since X is compact, f (X ) is compact. The collection
is an open cover of f (X ), hence has a finite subcover
B f (x), 3
x ∈X
B f (x 1 ), , . . . B f (x n ),
.
3
3
Define compact subsets {K1 , . . . , K n } of X and open subsets U1 , . . . ,Un of Y by
Ki := f −1 B f (x i ),
and Ui := B f (x i ), .
3
2
Since f is continuous, f ( A) ⊂ f ( A) for any set A. So,
62
5 Adjunctions and the compact open topology
⊂ B f (x i ),
= Ui
f (Ki ) ⊂ B f (x i ),
3
2
n S(K ,U ). To see
for each i = 1, . . . , n. Therefore, f is in the open set O := ∩i=
i
i
1
that O ⊂ B( f , ), let g ∈ O. If x ∈ Ki for some i, we have f (x), g(x) ∈ Ui since
f , g ∈ S(Ki ,Ui ). Therefore,
d( f (x), g(x)) ≤ d( f (x), f (x i )) + d( f (x i ), g(x i )) =
+ = .
2 2
)
( Since the balls B f (x i ), 3 cover f (X ), the compact sets {Ki } cover X and every
point x lies in Ki for some i. Therefore, d( f (x), g(x)) < for every x ∈ X and so
d( f , g) < in hom(X,Y ).
To show that the metric topology is finer than the compact-open topology, let
K ⊂ X be compact, U ⊂ Y be open and consider f ∈ S(K,U). From Lemma
5.1, we know there exists a fixed > 0 so that for any y ∈ f (K ) and any y 0 ∈
Y \ f (U), d(y, y 0 ) ≥ . Then, if g ∈ B( f , ), we have d( f (x), g(x)) < for
every x ∈ X. Therefore, if x ∈ K, g(x) ∈ U and we see that g(K ) ⊂ U. This
proves B( f , ) ⊂ S(K,U). If O = S(K1 ,U1 ) ∩ · · · ∩ S(K n ,Un ) is any basic open
set in the compact-open topology, we have the open metric ball B( f , ) ⊂ O where
= min{ 1 , . . . , n }. This proves that every basic open set in the compact-open
topology is open in the metric topology and hence the metric-topology is finer than
the compact-open topology.
5.5.2 The theorems of Ascoli and Arzela
It’s often important in analysis to determine when a given set of functions A ⊂
hom(X,Y ) is compact in the compact-open topology. It’s not hard to decide when a
family of functions is compact using the product topology:
Theorem 5.3. Prove that if Y is Hausdorff, then a subset A ⊂ hom(X,Y ) is compact
in the product topology if and only if A is closed in the product topology and for each
x ∈ X, the set Ax = { f (x) ∈ Y : f ∈ A} has compact closure in Y .
Proof. Exercise.
So, if one can identify families of functions for which the product topology and
the compact-open topology coincide, then one has necessary and sufficient conditions for such families to be compact in the compact open topology. The following
definition helps in making such an identification.
Definition 5.4. Let X be a topological space and (Y, d) be a metric space. A family
F ⊂ hom(X,Y ) is called equicontinuous at x ∈ X if and only if for every > 0,
there exists an open neighborhood U of x so that for every y ∈ U and for every
f ∈ F , d( f (x), f (y)) < . If F is equicontinuous for every x ∈ X, the family F is
simply called equicontinuous.
5.6 The compact-open topology when X is locally compact Hausdorff
63
One fact about equicontinuous families is that the compact-open topology agrees
with the product topology on them.
Lemma 5.2. Let X be a topological space and (Y, d) be a metric space. If F ⊂
hom(X,Y ) is an equicontinuous family, then the compact-open topology agrees with
the product topology.
Proof. It suffices to show that if { f α } is a net in F and f α → f pointwise, then
f α → f in the compact open topology. And this is a good exercise.
A second fact about equicontinuous families is that their closure is also equicontinuous.
Lemma 5.3. If F ⊂ hom(X,Y ) is equicontinuous, then the closure of F in
hom(X,Y ) using the product topology is also equicontinuous.
Proof. Let { f α } be a net in F that converges to a function f .... The rest is an exercise
in the definition of equicontinuous.
Now we have the right context for Ascoli’s theorem.
Ascoli’s Theorem. Let X be locally compact and Hausdorff, and let (Y, d) be a
metric space. A family F ⊂ hom(X,Y ) has compact closure if and only if F is
equicontinuous and for every x ∈ X, the set Fx := { f (x) : f ∈ F } has compact
closure.
Proof. Here we outline the proof. To prove the “if” part, let G be the closure of F
in Y X in the product topology. By Lemma 5.3, we know that G is equicontinuous.
Then G is compact in Y X with the product topology, which by Lemma 5.2 implies
it’s compact in the compact open topology. Then, F is a closed set in the compact
set G, hence is compact.
To prove the “only if” part, let G be any compact set in hom(X,Y ) containing
the family F . One can show (See section 47 of [21] for details) that G (hence F )
is equicontinuous. The evaluation map is continuous so the sets Gx are compact
(hence the sets Fx have compact closure).
Here’s a famous corollary:
Arzela’s Theorem. Let X be compact, (Y, d) be a metric space, and { f n } be a sequence of functions in hom(X,Y ). If { f n } is equicontinuous and for each x ∈ X, the
set { f n (x)} is bounded, then { f n } has a subsequence that converges uniformly.
5.6 The compact-open topology when X is locally compact
Hausdorff
We now return to prove the claims at the beginning of this section about the
compact-open topology on hom(X,Y ) having desirable properties when X is locally
compact and Hausdorff. We’ll need a few lemmas about so-called normal spaces.
64
5 Adjunctions and the compact open topology
5.6.1 Lemmas about normal spaces
Definition 5.5. A topological space X is called normal if and only if X is T1 and for
every pair of disjoint closed set C and D there exist disjoint open sets U and V with
C ⊂ U and D ⊂ V .
Theorem 5.4. Every compact Hausdorff space is normal.
Proof. The proof is a slight variation of the proof of Theorem 2.16.
Lemma 5.4. Let X be normal and U ⊂ X be open. For every closed set C with
C ⊂ U, there exists an open set V with C ⊂ V ⊂ V ⊂ U.
Proof. For any closed C ⊂ U, the sets C and X \ U are closed so there exist disjoint
open sets V and W with C ⊂ V and X \ U ⊂ W . Then, C ⊂ V ⊂ D ⊂ U where
D = X \ W . Since V is the smallest closed set containing V , we have V ⊂ D ⊂ U,
giving the result.
A corollary of Lemma 5.4 is that open covers of normal spaces have shrinkings.
Corollary 5.1. Let X be normal and suppose that {U1 , . . . ,Un } is an open cover of
X. There exists a shrinking of this cover. That is, there exists another open cover
{V1 , . . . ,Vn } of X with Vi ⊂ Vi ⊂ Ui .
And here’s the tube lemma:
Lemma 5.5. For any open set U ⊂ X × Y and any set K × {y} ⊂ U with K ⊂ X
compact, there exist open sets V ⊂ X and W ⊂ Y with K × {y} ⊂ V × W ⊂ U.
Proof. For each point (x, y) ∈ K × {y}, there is are open sets Vx ⊂ X and W x ⊂ Y
with (x, y) ⊂ Vx × W x ⊂ U. Then, {Vx } x ∈K is an open cover of K; take a finite
subcover {V1 , . . . ,Vn }. Then V = V1 ∪ · · · ∪ Vn and W = W1 ∩ · · · ∩ Wn are open
sets with K × {y} ⊂ V × W ⊂ U.
5.6.2 The compact open topology is exponential
First, the compact open topology is sufficiently coarse so that the map
hom(X × Z,Y ) → hom(Z, hom(X,Y ))
g 7→ g
(5.3)
t
is defined for all spaces X, Y , and Z.
Theorem 5.5. For any topological spaces X,Y and Z, if g : X ×Z → Y is continuous
then the set-theoretic adjoint g t : Z → hom(X,Y ) is continuous.
5.6 The compact-open topology when X is locally compact Hausdorff
65
Proof. Suppose g : X × Z → Y is continuous. To show that g t is continuous, consider a sub-basic open set S(K,U) in hom(X,Y ). We need to show that
(g t ) −1 (S(K,U)) = {z ∈ Z : f (K, z) ⊂ U } is open in Z. Let z ∈ (g t ) −1 (S(K,U)).
So, z ∈ Z and g(K, z) ⊂ U. Since g is continuous, we know that g −1 (U) =
{(x, z) : f (x, z) ⊂ U} is open in X × Z and contains K × {z}. Therefore, the
tube lemma 5.5 says there are open sets V and W with K ⊂ V and z ∈ W with
K × {z} ⊂ V × W ⊂ g −1 (U). Then, z ∈ W ⊂ (g t ) −1 (S(K,U)) as needed.
Theorem 5.5 proves the function in (5.3) is defined. Set theoretically, this map
is injective. For locally compact Hausdorff spaces X it is also surjective. This is
e
equivalent to proving that the evaluation map X × Y X −→ Y is continuous, which
isn’t hard to check for X locally compact and Hausdorff.
Lemma 5.6. Suppose hom(X,Y ) has a topology for which the evaluation map is
continuous. A function g : X × Z → Y is continuous if its adjoint g t : Z →
hom(X,Y ) is continuous. Conversely, if hom(X,Y ) has a topology for which the
continuity of a transpose g t : Z → hom(X,Y ) implies the continuity of g : X × Z →
Y , then the evaluation map is continuous.
Proof. The proof is mapped out by understanding that the evaluation map is the
counit of the ×−hom adjunction in Set. Here’s the argument: Consider the following
diagram
id ×g t
→ X × hom(X,Y )
←
←
X×Z
e
→Y
If g t is continuous, the composition g = e ◦ (id ×g t ) is continuous since the evaluation map e is continuous.
To prove the other way, assume hom(X,Y ) has a topology for which the continuity of a transpose g t : Z → hom(X,Y ) implies the continuity of g : X × Z → Y .
Since the transpose of the evaluation map e t : hom(X,Y ) → hom(X,Y ) is the identity, hence continuous, the hypothesis implies the evaluation map e is continuous.
Lemma 5.7. If X is locally compact and Hausdorff, then the evaluation map e :
X × hom(X,Y ) → Y is continuous.
Proof. We’ll prove that e is continuous at every point (x, g). Let (x, g) ∈ hom(X,Y )×
X and let U ⊂ Y be an open set containing ev(x, g) = g(x). Then g −1 (U) is an open
set in X containing x. Since X is locally compact and Hausdorff, there exists an
open set V ⊂ X with K := V compact and x ∈ V ⊂ K ⊂ g −1 (U). This implies
that g(x) ∈ f (K ) ⊂ U. Then V × S(K,U) is an open set in X × hom(X,Y ) with
(x, g) ∈ V × S(K,U) and ev(S(K,U) × V ) ⊂ U.
This proves the surjectivity, hence bijectivity of (5.3). Thus, for X locally compact, the compact open topology on hom(X,Y ) is exponential and we can denote it
by Y X . Summarizing, we have the following theorem.
Theorem 5.6. Suppose X is locally compact and Hausdorff. Then, the compact open
topology on hom(X,Y ) is exponential and we denote it by Y X . Then, for any spaces
Y and Z, we have an isomorphism of sets
66
5 Adjunctions and the compact open topology
hom(X × Z,Y ) −
→ hom(Z,Y X ).
(5.4)
5.6.3 Enrich the product-hom adjunction in Top
When is the bijection of sets in Equation (5.4) is continuous? An answer to this
question is: when X is locally compact and Hausdorff and Z is Hausdorff, then the
map hom(X × Z,Y ) → hom(Z, hom(X,Y )) is a homeomorphism. To prove it, it
will be convenient to use a smaller sub-basis for the compact-open topology that is
available when the domain of a mapping space is Hausdorff—in this case, it turns
out that the sets {S(K,U)} where U only ranges over a sub-basis of the topology of
the target space is still a sub-basis for the compact open topology.
Lemma 5.8. If X is Hausdorff and S is a sub-basis for the topology on Y , then the
sets {S(K,U) : K ⊂ X is compact and U ∈ S} are a sub-basis for the compact-open
topology on hom(X,Y )
Proof. Let f ∈ S(K,U) for some open set U ⊂ Y . Write U = ∪Uα and each Uα as
Uα = Vα, 1 ∩ · · · ∩ Vα, n α .
The collection { f −1 (Uα )} is an open cover of K. Let { f −1 (U1 ), . . . , f −1 (Uk )} be a
finite subcover. Since K is compact and Hausdorff, it is normal. Therefore, there is
a shrinking of the cover. Call the shrinking {V1 , . . . ,Vk }. For each i = 1, . . . , k let
Ki := Vi . Then K1 , . . . , Kk is a cover of K by compact sets with the property that
Ki ⊂ f −1 (Ui ) ⇒ f (Ki ) ⊂ Ui . Then for each i = 1, . . . , k,
ni
\
f ∈ S(Ki ,Ui ) = S Ki , ∩nj=i 1 Vi, j =
S(Ki ,Vi, j )
j=1
hence f ∈
T
i, j
S(Ki ,Vi, j ). This proves the lemma.
Theorem 5.7. If X is locally compact and Hausdorff and Z is Hausdorff, then for
any space X, the set map hom(X × Z,Y ) → hom(Z,Y X ) is a homeomorphsim.
Proof. First, we show that the collection
{S( A × B,U) : A ⊂ X, B ⊂ Z are compact and U ⊂ Y is open}
is a sub-basis for hom(X × Z,Y )). Let f ∈ S(K,U) for some compact set K ⊂ X × Z.
Then K ⊂ f −1 (U) and for each p = (x, z) ∈ K, we have open sets Vp ⊂ X and
W p ⊂ Z with p ∈ V × W ⊂ f −1 (U). By compactness, K is covered by finitely many
{Vi × Wi }. Now, let K 0 = π1 (K ) and K 00 = π2 (K ) be the projections of K onto X
and Z respectively. Since K 0 and K 00 are compact and Hausdorff, they’re normal.
So, there are shrinkings {Vi0 } and {Wi0 } that cover K 0 and K 00 respectively. Then Vi0
5.7 Compactly generated weakly Hausdorff spaces
67
and Wi0 are compact sets in X and Z with K ⊂ ∪i Vi0 × Wi0 ⊂ f −1 (U). Therefore,
f ∈ ∩i S(Vi0 × Wi0,U).
Since S = {S( A,U)} is a sub-basis for the compact open topology on hom(X,Y )
and Z is assumed to be Hausdorff, Lemma 5.8 says that
{S(B, S( A,U)) : A ⊂ X, B ⊂ Z are compact and U ⊂ Y is open}
is a sub-basis for hom(Z, hom(X,Y )). Therefore, the theorem follows from observing that the image of (S( A × B,U)) = S(B, S( A,U)).
Now, let’s make some concluding remarks about the compact open topology on
Y X when X is locally compact Hausdorff. From a certain point of view, it’s insufficient to work with locally compact Hausdorff spaces. For example, these spaces are
not closed under many common contructions. The colimit of the following diagram
of locally compact Hausdorff spaces
R ,→ R2 ,→ R3 ,→ · · ·
is not locally compact. Moreover, even if X is locally compact and Hausdorff, Y X
with the compact open topology may not be locally compact and Hausdorff, so the
construction of a topology on a mapping space is not repeatable. One solution to
all these issues involves more adjunctions. Specifically, adjunctions involving kification and “weak-Hausdorffification."
5.7 Compactly generated weakly Hausdorff spaces
Here, we present a bird’s eye view of constructing a topology on hom(X,Y ). The
main idea is to find a “convenient category” of topological spaces which has limits
and colimits, has exponential objects with the desirable properties listed in Section
5.5, and is large enough to contain the spaces we care about. Such a category is
the category of compactly generated weakly Hausdorff spaces. The importance of
compactly generated spaces for topologies on function spaces was recognized early
on [3]. The categorical perspective along the lines we present here, including the
behavior of compactly generated spaces under limits, occured later [25]. Compactly
generated Hausdorff spaces are a good setting to discuss function spaces and limits, but the Hausdorff property isn’t preserved by colimits. Finally, Hausdorff was
replaced with weakly Hausdorff yielding a convenient category [18].
The reader should be aware that the terminology is not used consistently in the
literature. Here we over-adjectivize our terminology to avoid any confusion.
Definition 5.6. A space X is compactly generated if and only if for all compact
spaces K and continuous maps f : K → X the set f −1 ( A) being closed (or open)
implies A is closed (or open).
68
5 Adjunctions and the compact open topology
Definition 5.7. A space X is weakly Hausdorff if and only if for all compact spaces
K and continuous maps f : K → X, the image f (K ) is closed in X.
Definition 5.8. Let CG, WH, and CGWH denote the full subcategories of compactly
generated, weakly Hausdorff, and compactly generated weakly Hausdorff spaces.
Example 5.4. The category CG includes all locally compact spaces and all first
countable spaces. The category CGWH includes locally compact Hausdorff spaces
and metric spaces. Notice that weakly Hausdorff lies between T1 and T2 . The interested reader will find these properties are all distinct by searching for examples
[24]
Given a space X, there is a smallest compactly generated topology on X containing the given topology. It is defined to consist of all sets U so that f −1 (U) is open
for any continuous function f : K → X from a compact space K that is continuous. The set X with this compactly generated topology is called the k-ification of
X and denoted by k X. The k-ification of a map f : X → Y is the map f viewed
as a map k X → kY , which will be continuous. Thus, k-ification defines a functor
k : Top → CG.
Theorem 5.8. The setup U : CG Top : k where U is the inclusion of CG → Top
and k is the k-ification functor is an adjunction.
Proof. Proved as Theorem 3.2 in [25].
Theorem 5.1 then implies that k preserves limits and U preserves colimits. The
statement that k preserves limits implies that the limit of a diagram in Top is sent by
k to the limit of the k-ification of the diagram. To clarify what this means, consider
two compactly generated spaces X and Y . The product X × Y in Top may not be
compactly generated, but k (X × Y ) is the product of X and Y in CG meaning that
it satisfies the universal property to be a product for compactly generated spaces. In
general, the topology on k (X × Y ) is finer than the topology on X × Y .
Now, the consequences of U being a left adjoint means that U preserves colimits.
This means that if a diagram in CG has a colimit in CG it must agree with the colimit
of the diagram in Top. While there’s no obvious reason that colimits in CG need
exist, it is nevertheless true.
Theorem 5.9. CG is closed under colimits.
Proof. See appendix A of [14].
Now, let’s add weakly Hausdorf to the picture.
Theorem 5.10. There is an adjunction q : CG CGWH : U where U is the inclusion of CGWH → CG.
Proof. See appendix A of [14].
5.7 Compactly generated weakly Hausdorff spaces
69
As a consequence, take a diagram in CGWH. The colimit of this diagram may
not be weakly Hausdorff, but it is compactly generated by Theorem 5.11. Then,
apply the functor q, which as a left adjoint preserves colimits, hence yields a space
in CGWH that must be the colimit of the diagram in CGWH.
As for limits, if a diagram in CGWH has a limit in CGWH it must agree with the
limit of the diagram in CG. While there’s no obvious reason that limits in CGWH
need exist, it is nevertheless true.
Theorem 5.11. CGWH is closed under limits.
Proof. See proposition 2.22 in [26].
The upshot of this back and forth game between adjoint functors produces a
category CGWH that is closed under limits and colimits. Be careful of multiple
interpretations. For example, for two compactly generated weakly Hausdorff spaces
X and Y , we have the “old” product, denote it X ×o Y which is the product in Top.
We also have the new product, denote it X × Y which is the product in CGWH.
Now, for compactly generated weakly Hausdorff spaces X and Y , let Y X =
k hom(X,Y ). That is, the topology we put on the space of maps from X to Y is
the k-ification of the compact open topology.
Theorem 5.12. If X and Y are CGWH then Y X is in CGWH. For a fixed X, the
assignment Y 7→ Y X defines a functor − X : CGWH → CGWH that fits into the
adjunction
X × −: CGWH CGWH :− X
inducing homeomorphisms of spaces hom(X × Z,Y ) hom(Z,Y X ).
Proof. See [14]
Corollary 5.2. For X,Y, Z ∈ CGWH we have
1. The functor − × X preserves colimits.
2. The functor − X preserves limits.
3. The functor Y − takes colimits to limits.
4. Composition Z Y × Y X → Z X is continuous.
5. Evaluation e : X × X Y → Y is continuous.
Moreover, CGWH has other good properties. Here are some examples, see the
excellent notes [26] for proofs:
•
•
•
•
•
The product of two quotient maps is the quotient of the product in CGWH.
Monomorphisms are precisely injections.
Epimorphisms are precisely maps with dense image.
Equalizers are precisely closed injections.
Coequlaizers are precisely quotient maps.
70
5 Adjunctions and the compact open topology
Exercises
5.1. Give examples and justify your answers.
(a) Find a space X and a space Y for which the evaluation map hom(X,Y ) × X → Y
is not continuous.
(b) Find a space X and a space Y for which the evaluation map Y X × X → Y is not
continuous.
5.2. F = { f a : 0 < a ≤ 1} where f a (x) = 1 − ax . Prove or disprove: F is compact
(in the compact-open topology).
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