Download Topology I with a categorical perspective

Tai-Danae Bradley and John Terilla
Topology I
with a categorical perspective
September 18, 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvi
0.3.1 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii
0.3.2 The emptyset and one point set . . . . . . . . . . . . . . . . . . . . . . . . . xvii
0.3.3 Products and coproducts in Set . . . . . . . . . . . . . . . . . . . . . . . . . xviii
0.3.4 Products and coproducts in any category . . . . . . . . . . . . . . . . . xix
0.3.5 Exponentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xx
0.4 Partially ordered sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xx
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xx
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 . . . . . . . . . .
1.4.3 Homotopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.5 Coproducts: two characterizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Connectedness and compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1 Connectedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.1 Definitions, theorems, and examples . . . . . . . . . . . . . . . . . . . .
2.1.2 The functor ⇡0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.3 Constructions and connectedness . . . . . . . . . . . . . . . . . . . . . . .
2.1.4 Locally (path) connected . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Hausdor↵ spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.1 Definitions, theorems and examples . . . . . . . . . . . . . . . . . . . . .
2.3.2 Constructions and compactness . . . . . . . . . . . . . . . . . . . . . . . .
2.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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17
17
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3
Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1 Closure and interior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 Nets and three theorems about them . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4 Categorical limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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4
Tychono↵’s Theorem (optional) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1 Preliminaries from set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.1 Axiom of choice and Zorn’s lemma . . . . . . . . . . . . . . . . . . . . .
4.2 Nets and compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3 A proof of Tychono↵’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4 Tychono↵’s theorem implies the axiom of choice . . . . . . . . . . . . . . . .
4.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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5
Adjoint functors and the compact open topology . . . . . . . . . . . . . . . . . . .
5.1 Example: The Stone-Čech compactification . . . . . . . . . . . . . . . . . . . . .
5.2 The unit and counit of an adjunction . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2.1 The unit of the Stone-Čech compactification . . . . . . . . . . . . .
5.3 Free-Forgetful adjunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4 The compact open topology: bird’s eye view . . . . . . . . . . . . . . . . . . . .
5.5 Some useful facts about compact sets . . . . . . . . . . . . . . . . . . . . . . . . . .
5.6 The compact open topology: frog’s eye view . . . . . . . . . . . . . . . . . . . .
5.6.1 The product-hom adjunction in Top . . . . . . . . . . . . . . . . . . . . .
5.6.2 Implications in homotopy theory . . . . . . . . . . . . . . . . . . . . . . .
5.6.3 Suspension-Loop adjunction . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.6.3.1 Wedge and smash . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.6.4 Enrich the adjunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.7 The theorems of Ascoli and Arzela . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.8 Universal properties and adjunctions (Optional) . . . . . . . . . . . . . . . . .
5.8.1 Corepresentable functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.8.2 Free groups again. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.8.3 Universal properties and adjunctions . . . . . . . . . . . . . . . . . . . .
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5.9
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
6
The Fundamental Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.1 Homotopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2 The fundamental group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3 Some applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3.1 Brouwer’s Fixed Point Theorem . . . . . . . . . . . . . . . . . . . . . . . .
6.3.2 Borsuk-Ulam Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3.3 The Ham Sandwich Theorem . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3.4 The Fundamental Theorem of Algebra . . . . . . . . . . . . . . . . . .
6.3.5 “Cylinder/Möbius question" from Spring 2016 qual . . . . . . .
6.4 The fundamental groupoid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Covering spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.1.1 Covering Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.1.2 The general lifting problem . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.1.3 Morphisms of covering spaces . . . . . . . . . . . . . . . . . . . . . . . . .
7.1.4 The Galois group of a covering . . . . . . . . . . . . . . . . . . . . . . . . .
7.2 The structure of the Galois group of a covering . . . . . . . . . . . . . . . . . .
7.2.1 Group actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2.2 The action of ⇡1 (X, x) on the fibers p 1 (x) of a cover . . . . . .
7.3 The existence of a universal cover . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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 whatever point-set
ideas might show up 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 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 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 ⌧ of subsets
of X, called open sets, satisfying the following properties:
• The sets ; and X are in ⌧.
• Any union of elements in ⌧ is also in ⌧.
• Any finite intersection of elements in ⌧ is also in ⌧.
The collection ⌧ 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 ⌧ ✓ ⌧ 0, the
topology ⌧ can be called coarser than ⌧ 0, or the topology ⌧ 0 can be called finer than
⌧. 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 co↵ee. A coarse grind yields a small number of chunky co↵ee pieces, whereas
a fine grind results in a large number of tiny co↵ee pieces. Finely ground beans
make stronger co↵ee, coursely ground beans make weaker co↵ee.
In practice, it can be convenient to work with a small collection of open subsets
of X that generates the topology.
xi
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0 Preliminaries
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 2 X there is a B 2 B such that x 2 B.
• If x 2 B1 \ B2 where B1 , B2 2 B, then there is at least one B3 2 B such that
x 2 B3 ✓ B1 \ B2 .
The topology ⌧ generated by the basis B is defined to be the smallest topology
containing B. Equivalently, a set U ✓ X is open in the topology generated the basis
B if and only if for every x 2 U there is a B 2 B such that x 2 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 2 X,
d(x, y) = d(y, x) for all x, y 2 X,
d(x, y) + d(y, z)  d(x, z) for all x, y, z 2 X
d(x, y) = 0 if and only if x = y for all x, y 2 X.
The function d is called a metric or a distance function. If (X, d) is a metric space,
x 2 X, and r > 0, the ball centered at x of radius r is defined to be
B(x,r) = {y 2 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 n-sphere S n := {x 2 Rn+1 :
d(x, 0) = 1}, and the closed unit ball D n := {(x 1 , . . . , x n ) 2 Rn : x 21 + · · · + x 2n  1}
are important topological spaces. The notation I := [0, 1], for interval, is often used
instead of D 1 .
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 (locally small) 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 set hom(X,Y ) called morphisms. The expression f : X ! Y means f 2 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
X
h
!Y
g
!
! Z
fg
f
! W
!
• 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, composition is composition of homomorphisms.
• Let G be a group. Define a category C with one object ⇤ and with mor(⇤, ⇤) = 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 base points, 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.
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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 2
homCop (X,Y ) and g 2 homCop (Y, Z ), the composition is f g 2 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 T op are also called homeomorphisms. Two sets X and Y that
are isomorphic are said to have the same cardinality. 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 property P and Y is
isomorphic to X, then Y has 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, Hausdor↵, 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.
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.
X
f
hom(Z, X )
f⇤
!Y
! hom(Z,Y )
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 .
0.2 Basic category theory
xv
f
X
!
hom(X, Z )
f⇤
!Y
hom(Y, Z )
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. This is an instance of the maxim that objects are determined by their
relationships with other objects.
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 ⇤ .
• 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 is discussed in detail in Chapter 6.
• 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 .
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 ).
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0 Preliminaries
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 ⌘ from F to G assigns a morphism ⌘ 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, ⌘Y F ( f ) = G( f )⌘ X . Here’s the picture
FX
! FY
GX
!
⌘Y
!
⌘X
F(f )
! GY
G( f )
For any two functors F, G : C ! D, let N at(F, G) denote the natural transformations
from F to G.
op
For any category C, there is a category SetC whose objects are functors from
Cop 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’s cartesian closed, it forms
what’s called a topos. We won’t dwell upon these properties further here, but here’s
one result that know about.
op
The Yoneda Lemma. For every object X in C and for every functor F 2 SetC , the
set of natural transformations from F to h X are isomorphic to F (X ).
There are many corollaries of the Yoneda lemma. Setting the functor F = hY in the
Yoneda lemma shows that N at(hY , h X ) hY (X ) = hom(Y, X ) and illustrates one
significance of the Yoneda lemma. The assignment X 7! h X defines a fully faithful
op
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 a very
op
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 Basic set theory
xvii
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 functions 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 .
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 : Z ! Y with g1 f = g2 f it
follows that g1 = g2 . That is, f is surjective if and only if f ⇤ : hom(Z,Y ) !
hom(Z, X ) is injective for all Z. Right-cancellative functions 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 (it needs to be checked
that having a left inverse and a right inverse both imply there’s a single two sided
inverse). Left-cancellative and right-cancellative together does 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.
xviii
0 Preliminaries
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
⇡1 : X ⇥ Y ! X and ⇡2 : 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 ⇡1 h = f 1 and ⇡2 h = f 2 . Here’s a picture
Z
h
X ⇥Y
!
⇡1
⇡2
!
!
!
X
f2
!
f1
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
i1
X
`
Y
!
X
!
i2
Y
h
Z
!
!
!
f1
f2
!
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 {X↵ }↵ 2 A is a set ↵ 2 A X↵ together with maps
`
i ↵ : X↵ ! ↵ 2 A X↵ satisfying the property that for any set Z and any collection of
`
functions { f ↵ : X↵ ! Z }, there is a unique map h : ↵ 2I X↵ ! Z with hi ↵ = f ↵
for all ↵ 2 A.
`
↵ 2 A X↵
h
i↵
X↵
f↵
!
! Z
0.3 Basic set theory
xix
The product of a collection of sets {X↵ }↵ 2 A is sometimes described as the subset
`
of functions f : A ! X↵ satisfying f (↵) 2 X↵ . 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 {X↵ }↵ 2 A is a set ↵ 2 A X↵ together with maps ⇡↵ : ↵ 2 A X↵ !
X↵ characterized by the property that for any set Z and any collection of functions
Q
{ f ↵ : Z ! X↵ }, there is a unique map h : Z ! ↵ 2 A X↵ with ⇡↵ h = f ↵ for all
↵ 2 A.
!
h
Z
f↵
↵2A
X↵
⇡↵
!
Q
! X↵
0.3.4 Products and coproducts in any category
Products and coproducts can be defined in any category using the universal properties described above. 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 are things to say something
about the existence of products and coproducts. The axiom of choice is precisely the
Q
statement that for any nonempty collection of sets {X↵ }↵ 2 A , the product ↵ 2 A X↵
exists and is nonempty. Coproducts in Set are also axiomatically guaranteed to exist,
by the axioms of union and extension.
In any category with products and coproducts, the universal properties of products and coproducts lead to the bijections
a
hom *
X↵ , Z +
, ↵
-
Y
↵
hom (X↵ , Z ) and hom * Z,
,
Y
↵
X↵ +
-
Y
hom (Z, X↵ ) .
↵
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
X↵
(X↵ ) ⇤
xx
0 Preliminaries
Remember, the dual of the sum is the product of the duals.
0.3.5 Exponentiation
The set 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
XZ ⇥YZ
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
(X Y ) Z X Z ⇥Y
which is an expression of the hom-⇥ adjunction. Fix a set Y . Let F be the functor
⇥ Y and let G be the functor G : hom( ,Y ). In this notation, (X Y ) Z
X Z ⇥Y
becomes hom(F (Z ), X )
hom(Z, GX ) and evokes of the defining property of
adjoint linear maps.
0.4 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 2 P, a  a;
transitive means that for all a, b, c 2 P, if a  b and b  c then a  c; antisymmetric
means that for all a, b 2 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 ⌧ containing S and that the collection of all finite intersections
of sets in S is a basis for ⌧. In this situation, the collection S is called a sub-basis
for the topology ⌧.
0.4 Partially ordered sets
xxi
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. Prove that if f : X ! Y is left-invertible then it is monic and if it is rightinvertible it is epic.
0.4. Give an example of a morphism that is epic but not right invertible.
0.5. Prove that if a morphism is left-invertible and right-invertible then it is invertible.
0.6. Give an example of a morphism that is epic and monic but not an isomorphism.
0.7. Discuss the initial object, the terminal object, products, and coproducts in the
categories Group and Vect.
0.8. Prove that f : X ! Y is an isomorphism if and only if f ⇤ : hom(Y, Z ) !
hom(X, Z ) is an isomorphism if and only if f ⇤ : hom(Z, X ) ! hom(Z,Y ) is an
isomorphism.
0.9. 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 ⌘ : h X ! F, one obtains a
special element ⌘(id X ) 2 F (X ) which completely determines ⌘.
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, products, coproducts, and quotients. 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 2 X : a < x < b} along with the
intervals (a, 1) and ( 1, 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 2 N}
for a 2 Z \ {0} and b 2 Z, together with ;, are open. Furstenberg [4] 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 2 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
• k↵vk = |↵|kvk for all scalars ↵ 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 =
1, the pi=1 |x i | . More generally, for any p
n
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 )k1 := sup{|x 1 |, . . . , |x n |}.
These norms define di↵erent metrics, with di↵erent 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 1
n=1 x n is finite is a subspace
of RN . Then l p with
1
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 di↵erenti p since the underlying sets are di↵erent. For instance, {1/n} is in l 2 but
not l 1 . Nonetheless, as topological spaces, the spaces l p are homeomorphic [8]. The
the set l 1 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.
1.1.2 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
U : X ! S defined by
8
>
< 1 if x 2 U,
U (x) = >
: 0 if x < U
is a continuous function, and every continuous function from X ! S is of the
form U 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 2 X is the same as a map ⇤ ! X, the points of a space X can
be found in 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 : [0, 1] ! X. A loop
in a space X is a continuous function : [0, 1] ! X with (0) = (1).
4
1 Examples and constructions
Example 1.12. If (X, d) is a metric space and x 2 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, ⌧discrete ) ! (R, ⌧usual )
is a continuous bijection that is not a homeomorphism. As we will see later, in the
subcategory of compact Hausdor↵ spaces, continuous bijections are always homeomorphisms.
1.2 The subspace topology
The subspace topology is often defined (for example, in [16]) as follows:
Definition 1.1. Let (X, ⌧X ) be a topological space and let Y be any subset of X. The
subspace topology on Y is defined by {U \ Y : U 2 ⌧X }.
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, ⌧X ) 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 ⌧f , that is the coarsest topology for
which ⌧ 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 ⌧f .
Note that ⌧f has a simple explicit description as ⌧f = { f 1 (U) : U ✓ X is open}.
One sees that the subspace topology on a subset Y ✓ X is the same as ⌧i where
i : Y ! X is the natural inclusion.
1.2 The subspace topology
5
Better Definition. Let (X, ⌧X ) 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 ⌧f , 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 ⌧f 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
coursest topology ⌧f 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 ⌧f induced by f .
Example 1.14. Consider the set [0, 1] with the discrete topology. The map i :
([0, 1], ⌧di scr et e ) ! (R, ⌧or dinar y ) is continuous, 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 ⌧Y determines the continuous maps to Y .
• The continuous maps to Y determine the subspace topology ⌧Y .
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, ⌧X ) 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, ⌧Z )
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
Z
i
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 ⌧Y be the subspace topology on Y . Let (Z, ⌧Z ) 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 ⌧ 0 is a topology on Y and that ⌧ 0 has the universal property.
We have to prove that this topology ⌧ 0 equals the subspace topology ⌧Y . We are
assuming that when Y has the topology ⌧ 0, then for every topological space (Z, ⌧Z )
and for any function f : Z ! Y , f is continuous if and only if i f is continuous. In
particular, if we let (Z, ⌧Z ) be (Y, ⌧Y ) where ⌧Y 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, ⌧Y )
idY
i
(Y, ⌧ 0 )
Since we know the function i idY = i : Y ! X is continuous when Y has the
subspace topology ⌧, the universal property implies that idY : (Y, ⌧Y ) ! (Y, ⌧ 0 ) is
continuous. This implies that the subspace topology ⌧Y is finer than ⌧ 0; i.e. ⌧ 0 ✓ ⌧Y .
To show that ⌧Y ✓ ⌧ 0, let (Z, ⌧Z ) be (Y, ⌧ 0 ) and let f = idY : (Y, ⌧ 0 ) ! (Y, ⌧ 0 ). So
we have the following picture
X
i idY = i
(Y, ⌧ 0 )
idY
i
(Y, ⌧ 0 )
1.3 The quotient topology
7
Since idY is continuous, we must have i idY = i : Y ! X continuous. That is, ⌧ 0 is
a topology on Y for which the inclusion i : Y ! X is continuous. Since the subspace
topology ⌧Y is the coarsest topology on Y for which i : Y ! X is continuous, we
conclude that ⌧Y is coarser than ⌧ 0; i.e., ⌧Y ✓ ⌧ 0 . The conclusion is that ⌧ 0 = ⌧Y .
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 ⇡ : 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 ⇡ 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 ⇡ : X ⇣ S is surjective then the set S is isomorphic to X/⇠ where ⇠ is the equivalence relation whose
equivalence classes are the fibers of ⇡:
x ⇠ y , ⇡(x) = ⇡(y).
The map ⇡ conveniently provides the isomorphism
'
S ! X/⇠
s 7! ⇡ 1 (s).
Conversely, if ⇠ is an equivalence relation on X, the natural projection ⇡ : 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
⇡ : 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 ⇡, 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 ⇡ : S ,! X as being defined
defined on the set S, or on the subset ⇡(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 ⇡ : X ! S is continuous: saying U is open only if ⇡ 1 (U) makes
⇡ continuous and saying U is open whenever ⇡ 1 (U) is open makes the quotient
topology the finest possible topology for which ⇡ 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 ⇡ : X ⇣
S be surjective. The quotient topology on S is the finest topology for which ⇡ 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 di↵erence 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. These will be
Given a surjection ⇡ : 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 ⇡ are continuous functions
X ! Z.
Universal property for the quotient topology. Let X be a topological space, let S
be a set, and let ⇡ : X ! S be surjective. For every topological space Z and every
function f : S ! Z, f is continuous if and only if f ⇡ : X ! Z is continuous.
Here is the picture:
X
f
⇡
S
f
⇡
Z
Theorem 1.2. The quotient topology is determined by the universal property described above.
Proof. Exercise.
1.4 The product topology
9
The universal property of the quotient topology tells us precisely which functions
S ! Z from a quotient to a space Z are: they are continuous maps f : X ! Z that
are constant on the fibers of ⇡ : X ! S.
Example 1.16. The map ⇡ : [0, 1] ! S 1 defined by ⇡(t) = (cos(2⇡t), sin(2⇡t)) 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 ⇡. That is, continuous
functions S 1 ! Z are the same as paths : [0, 1] ! X satisfying (0) = (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 ' x for 2 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 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 {X↵ }↵ 2 A be an arbitrary collection of topological spaces and consider the set
Y
X=
X↵ .
↵2A
10
1 Examples and constructions
We’d like to make the set X into a topological space. Sometimes the product topology is defined by
Definition 1.4. The product topology on X is defined to be the topology generated
by the basis
Y
8
9
>
>
<
=.
U
:
U
✓
X
is
open,
and
all
but
finitely
many
U
=
X
↵
↵
↵
↵
↵
>
>
:↵ 2 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 ⇡↵ : X ! X↵ . Observe that there are topologies on X, such as the discrete topology, that make the projections ⇡↵ : X ! X↵
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 {X↵ }↵ 2 A be an arbitrary collection of topological spaces
Q
and let X = ↵ 2 A X↵ . The product topology on X is defined to be the coarsest
topology on X for which all of the projections ⇡↵ 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 {X↵ }↵ 2 A be an arbitrary collecQ
tion of topological spaces and consider the set X = ↵ 2 A X↵ . Keeping in mind
that the universal property of the disjoint union of sets says that set functions into X
are the same as collections of functions into the sets X↵ , it’s not hard to guess that
Q
Z ! X = ↵ 2 A X↵ is continuous whenever all the components Z ! X ! X↵ are
continuous.
Theorem 1.3. Let {X↵ }↵ 2 A be an arbitrary collection of topological spaces and
Q
let X = ↵ 2 A X↵ . Let ⇡↵ : X ! X↵ 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, ⌧Z )
and every function f : Z ! X, f is continuous if and only if for every ↵ 2 A,
the component f ⇡↵ : Z ! X↵ is continuous.
1.4 The product topology
11
Here is the picture:
X
f
Z
⇡↵
X↵
f↵
Proof. Exercise.
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
!
f
x
R
⇡1
⇡2
!
!
!
!
R
y
2
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
!
x7! (x, y 0 )
!
R2
f
!
!
S
f (x 0, y)
!
f (x, y 0 )
y7! (x 0, y)
For example, f : R2 ! R defined by
xy
8
>
>
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
1.4.3 Homotopy
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 is the category hTop
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 ' id X . Functors from hTop are called
homotopy invariants.
Example 1.20. For an example, R2 ' ⇤. Define f : ⇤ ! R2 by ⇤ 7! 0 and define
g : R2 ! ⇤ in the only way possible. Then g f = id⇤ and f g : R2 ! 0, 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 ↵, : [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) = ↵(t), h(t, 1) = (t), and h(0, s) = x and h(1, s) = y for all
s,t 2 [0, 1]. In other words, for all s, the path t 7! h(t, s) is a path from x to y,
agreeing with ↵ at s = 0 and at s = 1.
1.5 Coproducts: two characterizations
Let {X↵ } be a collection of topological spaces. We’d like to make the disjoint union
`
X into a topological space. It turns out, that the disjoint union of the topologies
` ↵
`
⌧↵ defines a topology on the X = X↵ . The construction seems so obvious that
it usually gets very little attention in the literature. What’s worth understanding is
what characterizes this topology, why this is the right topology? Set theoretically,
`
the disjoint union comes with functions X↵ !
X↵ and for a first definition,
we’d like a topology on X for which these inclusions are continuous. There are
`
many topologies on X↵ for which these inclusions are continuous— the indiscrete
1.6 Exercises
13
`
topology is one—the right topology to put on X↵ is the finest topology for which
`
the maps X↵ ! X↵ are continuous.
To characterize this topology a second way, remember that set theoretically, func`
tions from X are determined by collections of functions from X↵ . Let X =
X↵
and let i ↵ : X↵ ! 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, ⌧Z ) and
every function f : X ! Z, f is continuous if and only if for every ↵ 2 A, f i ↵ :
X↵ ! Z is continuous. Here is the picture:
!
X
f
i↵
X↵
f
!
Z
!
i
↵
Example 1.21. Any set X is the coproduct over its points of the one point set:
a
X'
{⇤}
As spaces, however X '
`
x 2X
x 2X {⇤}
if and only if X has the discrete topology.
1.6 Exercises
1.1. Draw a diagram of all the topologies on a three point set, indicating which are
contained in which.
1.2. Let (X, d) be a metric space. Show that for any x 2 X and any r > 0, the set
{y 2 X : d(x, y)  r } is closed. Give an example to show that
{y 2 X : d(x, y)  r } may not equal B(x,r).
1.3. 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.
1.4. 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].
1.5. Give an example of a path p : [0, 1] ! X connecting a to b in the space (X, ⌧)
where
X = {a, b, c, d}
and
⌧ = {;, {a}, {c}, {a, c}, {a, b, c}, {a, d, c}, X }.
14
1 Examples and constructions
1.6. Prove that any two norms on a finite dimensional vectors space (over R or C)
give rise to homeomorphic topological spaces.
1.7. Prove that l 1 is not homeomorphic to l p for p , 1.
1.8. Let C([0, 1]) denote the set of continuous functions on [0, 1]. The following
define norms on C([0, 1]):
k f k1 = sup | f (x)|.
x 2[0, 1]
k f k1 =
Z
1
0
| f |.
Prove that the topologies on C([0, 1]) coming from these two norms are di↵erent.
1.9. For your reference, we state The Cantor-Schroder-Bernstien Theorem for sets.
It’s stated and proved on page 28 of [9], in Section 3 of [11], etc...
The Cantor-Schroeder-Bernstein Theorem. Let X and Y be sets and let f : X ! Y
be injective and let g : Y ! X be injective. Then there exists a bijection h : X ! Y .
(a) Prove (or read the proof) of the Cantor-Schroeder-Bernstein theorem.
(b) Show that there can be no such theorem for topological spaces. That is, give an
example of two, non homeomorphic topological spaces X and Y , an embedding
f : X ! Y , and an embedding g : Y ! X.
Q
1.10. Prove Theorem 1.3. That is, prove that X = ↵ 2 A X↵ 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.
1.11. Are the subspace and product topologies are consistent with each other? Let
{X↵ }↵ 2 A be a collection of topological spaces and let {Y↵ } be a collection of subsets;
Q
each Y↵ ✓ X↵ . There are two things you can do to put a topology on Y = ↵ 2 A Y↵ :
1. You can take the subspace topology on each Y↵ , 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.
1.12. Prove that the quotient topology is characterized by the universal property
given in Section 1.3.
1.13. Are the quotient and product topologies are compatible with each other? Let
{X↵ }↵ 2⇤ be a collection of topological spaces, let {Y↵ }↵ 2⇤ be a collection of sets,
Q
and let {⇡↵ : X↵ ⇣ Y↵ }↵ 2⇤ be a collection of surjections. Let X = ↵ X↵ and
notice that you have a surjection ⇡ : X ⇣ Y . There are two ways to put a topology
Q
on Y = ↵ 2 A Y↵ :
1.6 Exercises
15
1. Take the quotient topology on each Y↵ , 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.
1.14. 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 ) 2 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.
R
ri
! X⇥X
⇡1
! X
⇡ 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 .
1.15. Consider R2 with the usual topology. Define an equivalence relation on R2 by
(x, y) ⇠ (x 0, y 0 ) , x y = x 0 y 0
and let Y := R2 / ⇠ denote the set of equivalence classes.
(a) Prove that, as a set, Y is isomorphic to R.
(b) Now consider Y as a topological space by equipping it with the quotient topology; i.e, the quotient topology induced by the natural projection
⇡ : R2 ! R2 / ⇠ .
Is Y homeomorphic to R?
1.16. 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, ⌧X ) and (Y, ⌧Y ) 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 ⌧Y on Y is equal to ⌧f ,
the quotient topology on Y .
1.17. Consider the closed disk D 2 and the two sphere S 2 :
D 2 = {(x, y) 2 R2 : x 2 + y 2  1}
S 2 = {(x, y, z) 2 R3 : x 2 + y 2 + z 2 = 1}.
16
1 Examples and constructions
Consider the equivalence relation on D 2 defined by identifying every point on the
S 1 ✓ D 2 . So each point in the 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,
Hausdor↵, 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 [21, 16, 9, 11].
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.
We like the second one. Recall that a path from x to y in a topological space X is a
map : I ! X with (0) = x and (1) = y.
Definition 2.2. A topological space X is said to be path connected if and only if for
all x, y 2 X there is a path that connects x and y.
There is an equivalence relation defined by x ⇠ y if and only if there is a path in
X connecting x and y. The equivalence classes of ⇠ are called the path components
17
18
2 Connectedness and compactness
of X and are denoted by ⇡0 (X ). Notice that the path components of X are homotopy
classes of maps ⇤ ! X: a point x 2 X is a map ⇤ ! X and a homotopy 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.
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
g
!
f
Y
g
!
! {0, 1}
Since Y is connected, g is constant, hence the composition g = g f is constant.
Theorem 2.3. Suppose X = [↵ 2 A X↵ and that X↵ is connected (or path connected)
for each ↵ 2 A. If there is a point x 2 \↵ 2 A X↵ then X is connected (or path
connected).
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).
(
p )
Example 2.1. The rational numbers Q are not connected since x 2 Q : x < 2
(
p )
and x 2 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.
2.1 Connectedness
19
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 2 A such that x < z < y for some z < A. Thus A = ( A \ ( 1, z)) [ ( A \
(z, 1)) 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 2 U and y 2 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 2 U
or s 2 V and so (s , s + ) ✓ U or (s , s + ) ✓ V for some > 0. If the former
holds, then s fails to be an upper bound on U 0. If the latter, then s
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 : [0, 1] ! [0, 1] has a fixed point.
Proof. Suppose f : [0, 1] ! [0, 1] is a continuous function for which f (x) , x for
all x 2 [0, 1]. In particular we have f (0) > 0 and f (1) < 1. Now define a map
g : [0, 1] ! { 1, 1} by
f (x) x
.
g(x) =
| f (x) x|
Then g is continuous and g(0) = 1 and g(1) = 1. But this is impossible since [0, 1]
is connected. (See item 3 of Definition 2.1 and note that all two-point sets with the
discrete topology are homeomorphic.)
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 6.
Here’s another nice result that follows from the connectedness of intervals. It is
due to Nandakumar and Rao (see [17] and [22]).
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 di↵erence of the
area on the left and the right. As the line moves from left to right the di↵erence 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
di↵erence between the perimeter on the left and the perimeter on the right. Rotate
20
2 Connectedness and compactness
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 ⇡0
Finally, the assignment X 7! ⇡0 (X ) is the object part of a functor ⇡0 : 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 ⇡0 f : A 7! the path component containing f ( A) defines a function from
⇡0 (X ) ! ⇡0 (Y ). To summarize the functor
⇡0 : Top ! Set
✓
X!
7 ⇡0 (X )
◆
✓
X !Y !
7
⇡0 (X )
f
⇡0 ( f )
! ⇡0 (Y )
◆
The fact that functors respect composition when applied to morphisms makes
them quite powerful. For example, here’s an alternative way to use the functor ⇡0 to
prove Theorem 2.6. Suppose f : [0, 1] ! [0, 1] is continuous. If f (x) , x for any x,
then the map g : [0, 1] ! {0, 1} defined by
g(x) =
1 x
2 |x
! 8
>0
f (x)
+1 = <
>1
f (x)|
:
if x < f (x),
if x > f (x).
is continuous. So, we have the diagram
id
{0, 1} -
i
! [0, 1]
!
g ! {0, 1}
Apply ⇡0 to get
⇡ 0 id=id
{0, 1} which is impossible.
! {⇤}
⇡0 i
!
⇡ 0 g! {0, 1}
2.1 Connectedness
21
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
2.10 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 {X↵ }↵ 2 A be a collection of connected (or path connected) topoQ
logical spaces. Then X = ↵ 2 A X↵ is connected (or path connected).
Proof. We’ll prove it for path connected and leave the connected part as an exercise.
Suppose that X↵ is path connected for every ↵ 2 A. Let a, b 2 X. Since each X↵
is path connected, there exists a path p↵ : [0, 1] ! X↵ connecting a↵ to b↵ . By
the universal property of the product topology, there exists a continuous function
p : [0, 1] ! X so that ⇡↵ p = p↵ . This function 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 {X↵ } 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=
X↵ .
However, whether X, as a topological space, is the coproduct of its connected components depends on whether the components X↵ are open. For example, the con`
nected components of the rationals Q consist of singletons {r }. As a set, Q = {r }
`
but certainly not as a topological space: r 2Q {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:
22
2 Connectedness and compactness
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 Locally (path) connected
Connected and path-connected have local versions.
Definition 2.3. A topological space is locally connected (or locally path connected)
if for each x 2 X and every neighborhood U ✓ X of x, there is a connected (or path
connected) neighborhood V of x with V ✓ U.
Theorem 2.10. If X is connected (or path connected) and f : X ! Y then f (X ) is
connected (or path connected).
Proof. Exercise.
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 2.11. 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."
Example 2.4. Let C = { n1 : n 2 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
2.3 Compactness
23
2.2 Hausdor↵ spaces
Now is a good time to define a topological property called Hausdor↵.
Definition 2.4. A space X is Hausdor↵ if and only if for every two points x and y,
there exist disjoint open sets U and V with x 2 U and y 2 V .
It’s not hard to prove that Hausdor↵ defines a topological property. Then, one
should look at which constructions preserve the property. One finds that subspaces
of Hausdor↵ spaces are Hausdor↵, products of Hausdor↵ spaces are Hausdor↵, coproducts of Hausdor↵ spaces are Hausdor↵, but quotients of Hausdor↵ spaces are
not-necessarily Hausdor↵. In fact, quotients of Hausdor↵ spaces are a great source
of non-Hausdor↵ spaces throughout the mathematical world.
Theorem 2.12. Every space X is the quotient of a Hausdor↵ space H.
Proof. Omitted. See [19].
Example 2.5. Metric spaces are Hausdor↵. To see this, let⇣ x and
⌘ y be points
⇣
⌘ in a
d
d
metric space. If x , y, then d := d(x, y) > 0. Then B x, 2 and B y, 2 are
disjoint open sets separating x and y.
Theorem 2.13. A space X is Hausdor↵ if and only the diagonal map
is closed.
: X ! X ⇥X
Proof. Exercise.
The Hausdor↵ property becomes more interesting when it’s mixed with other topological properties.
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 2.14. If X is compact and f : X ! Y then f (X ) is compact.
Proof. Exercise.
Corollary 2.2. Compactness is a topological property.
24
2 Connectedness and compactness
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
S 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 2 F, then there is an open set Ux with Ux \ F = {x}. Then {Ux } x 2X
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, 1) : x 2 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 2.15. A space X is compact if and only if every collection of closed subsets
of X with the FIP has nonempty intersection.
Proof. Exercise.
The reader expecting a more cateogrical definition of compactness will find a
suitable alternative definition in the following theorem of Bourbaki [?].
Theorem 2.16. X is compact if and only if for all spaces Y the projection X ⇥Y ! Y
is a closed map.
Proof. Exercise.
Theorem 2.17. Closed subsets of compact spaces are compact.
Proof. Let X be compact with C ✓ X closed and suppose U = {U↵ }↵ 2 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 Hausdor↵ spaces are quite nice—they can be separated from
points by open sets.
2.3 Compactness
25
Theorem 2.18. Let X be Hausdor↵. For any point x 2 X and any compact set
K ⇢ X \ {x} there exist disjoint open sets U and V with x 2 U and K ⇢ V .
Proof. Let x 2 X and let K ( X be compact. For each y 2 K, there are disjoint
open sets Uy and Vy with x 2 Uy and y 2 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 2 U and K ⇢ V .
Corollary 2.3. Compact subsets of Hausdor↵ spaces are closed.
Corollary 2.4. If X is compact and Y is Hausdor↵ 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 Hausdor↵ space and let
C ⇢ X be closed. Then C is compact, so f (C) is compact, so f (C) is closed.
Definition 2.7. A space X is locally compact if and only if for every point x 2 X
there exists a compact set K and an open set U with x 2 U ⇢ K.
Example 2.7. Every compact space is locally compact. Rn is locally compact. Every
discrete space is locally compact. The real line with the lower limit topology ⌧ll (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, ⌧di scr et e ) ! (R, ⌧ll ) gives an example. Nonetheless, locally
compact is a topological property.
Now, the product and quotient topologies are not compatible in the sense of Problem 1.13. The hypothesis of locally compact and Hausdor↵ makes the situation
much better.
Theorem 2.19. If X1 ⇣ Y1 and X2 ⇣ Y2 are quotient maps and Y1 and X2 are
locally compact and Hausdor↵, then X1 ⇥ X2 ⇣ Y1 ⇥ Y2 is a quotient map.
Proof. Exercise.
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
2.17 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. As for products, we have Tychono↵’s theorem:
Tychono↵’s Theorem. The product of compact spaces is compact.
Proof. In Chapter 4.
26
2 Connectedness and compactness
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 Hausdor↵ and all compact subsets of a Hausdor↵ space
must be closed, K is closed.
Conversely (and this is the part that uses the Tychono↵ 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., ⇡i (K ) ⇢ [ai , bi ] for each i. Then K ⇢ [a1 , b1 ] ⇥
[a2 , b2 ] ⇥ · · · ⇥ [an , bn ]. Since each set [ai , bi ] is compact, the Tychono↵ 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.
2.4 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 2 X there is an
open set U with x 2 U and f |U constant. Prove or disprove: if X 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 2 N and a, b 2 N are relatively prime}
Prove that N with this topology is connected. [5]
Q
Q
2.6. Let {X↵ } be a collection of spaces. Prove that ⇡0 ( X↵ )
⇡0 (X↵ ). The
special case ⇡0 (X↵ ) = ⇤ for all ↵ 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.
2.4 Exercises
27
2.8. Show that Q ✓ R with the subspace topology is not locally compact.
2.9. Define a space X to be pseudocompact i↵ 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 2 X, B(x, ✏ ) is contained in some U 2 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 of Chapter 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 2 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 } 2 l 2 : 1
n=1 x n  1 be the closed unit ball in l . Show that B
is not compact.
2.16. Show that the product of Hausdor↵ spaces is Hausdor↵. Give an example to
show that the quotient of a Hausdor↵ space need not be Hausdor↵.
2.17. For any map f : X ! Y , the set = {(x, y) 2 X ⇥ Y : y = f (x)} is called the
graph of f . Suppose now that X is any space and Y is compact Hausdor↵. Prove that
is closed if and only if f is continuous. Is the compactness condition necessary?
(This is called the closed graph theorem.)
2.18. Let X be a Hausdor↵ space with f : X ! Y a continuous closed surjection
such that f 1 (y) is compact for each y 2 Y . Prove that Y is Hausdor↵.
2.19. Prove or disprove: If f : X ! Y is a continuous bijection and X is Hausdor↵
then Y must be Hausdor↵.
2.20. 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.
(a) Prove that if A is dense in X and Y is Hausdor↵, 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.
28
2 Connectedness and compactness
(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.
2.21. Prove or disprove: X is Hausdor↵ if and only if
{(x, x, x, x, x, . . .) 2 X N : x 2 X }
is closed in X N .
(
)
2.22. Let K = n1 : n 2 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 the [0, 1] is not a compact subspace of R with the K-topology.
(b) More generally, prove that if ⌧ is any topology on [0, 1] finer than the ordinary
one, then [0, 1] cannot be compact in the topology ⌧.
(c) Prove that if ⌧ is any topology on [0, 1] courser than the usual one, then [0, 1]
cannot be Hausdor↵ in the topology ⌧.
Chapter 3
Limits
A comathematician is a device for turning cotheorems into ↵ee.
—?
3.1 Closure and interior
Definition 3.1. 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 open
set
⇣ ⌘contained in B. A set B is called dense if B = X and is called nowhere dense if
B = ;.
In general, there is no smallest open set containing B and there is no largest
closed set contained in B. This shouldn’t be surprising since largest and smallest
things don’t always exist and aren’t always well-defined, even when the meaning of
largest and smallest is clear: is there a largest integer? is there a largest element of
{{1, 2, 3}, {red, green, glue}, {a, b, c}} ?
First, the meaning of largest and smallest can be clarified. Saying that B is the
smallest closed set containing B means that if C is any closed set containing B then
B ✓ C. Saying that B is the largest open set containing B means that if O is any
open set contained in B then O ✓ B . To see that B exists and is unique, let ⇤ be
the set of all closed sets containing B. Note that ⇤ is nonempty since X 2 ⇤ and
\ D 2⇤ D is a closed set containing A. If C is a closed set containing B, then C 2 ⇤
therefore \ D 2⇤ D ✓ C. The existence of the interior is proved similarly.
3.2 Sequences
Definition 3.2. 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 }
29
30
3 Limits
converges to z 2 X if and only if for every open set U containing z, there exists an
N 2 N so that if n N, x n 2 U. If {x n } converges to z 2 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
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 ⌧ = {;, {1}, {1, 2}, A}. Then the constant sequence 1, 1, 1, 1, . . . converges to 1; it also converges to 2 and to 3. Observe
that A is not T1 : there is no open set around 1 separating it from 2.
Example 3.2. Consider Z with the cofinite topology. For any m 2 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 2 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 2 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 2 U and y 2 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 2 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 are useful tools that can be used to detect certain properties of spaces,
subsets of spaces, and functions between spaces. Here are a few theorems about
sequences.
Theorem 3.1. In a Hausdor↵ space, limits of sequences are unique.
Proof. Let X be Hausdor↵, 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 2 U and y 2 V. Since
{x n } ! x there is a number N so that x n 2 U for all n N. Since {x n } ! y there
is a number K so that x n 2 U for all n K. Let M = max{N, K }. Since M
N
and M K we have x M 2 U and x M 2 V contradicting the fact that U and V are
disjoint.
Theorem 3.2. 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 3.3. If {x n } is a sequence in A that converges to x, then x 2 A.
Proof. Exercise.
3.2 Sequences
31
Before continuing, let’s enrich the discussion by introducing a couple more topological properties—two of the so-called “separation” axioms.
Definition 3.3. Three fundamental separation axioms:
• A topological space X is T0 i↵ for every pair of points x, y 2 X there exists an
open set containing one, but not both of them.
• A topological space X is T1 i↵ for every pair of points x, y 2 X there exists open
sets U and V with x 2 U, y 2 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 2 X there exist open sets U and V with x 2 U,
y 2 V with U \ V = ; is sometimes called T2 , but we’ve already named the property
Hausdor↵, after Felix Hausdor↵ who originally used the axiom in his definition of
“neighborhood spaces." [7].
Theorem 3.4. If X is T1 , then for any x 2 X, the constant sequence x, x, x, . . .
converges to x and only to x.
Proof. Suppose X is T1 and x 2 X. It’s clear that x, x, x, . . . ! x. Let y , x. Then
there exists an open set U with y 2 U and x < U. Therefore, x, x, x, . . . cannot
converge to y.
Now, in fact, T1 spaces are characterized by the property that constant sequences
have only one limit.
Theorem 3.5. A topological space X is T1 if and only if for every point x, the constant sequence x, x, x, . . . converges to x and only to x.
Proof. Theorem 3.4 establishes the only if part. For the if part, suppose that X is not
T1 . Then, there exist two distinct points x and y for which every open set around y
contains x. So the sequence x, x, x, . . . ! y.
One can ask whether sequences suffice to detect Hausdor↵ spaces, continuous
functions, or closed sets. That is, whether Theorems 3.1, 3.2, and 3.3, have if and
only if versions. In general, the answer is no.
Example 3.4. Sequences don’t suffice to detect Hausdor↵ spaces. Consider R with
the cocountable topology. This space is not Hausdor↵, 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 .
32
3 Limits
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.
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 3.4. Let X be a space. A collection of open sets C is called a neighborhood base for x 2 X if for every open set O containing x, there exists an open set
U 2 C with x 2 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 base.
Example 3.7. An n-dimensional manifold is a second countable Hausdor↵ 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 sequences do suffice to characterize properties of spaces,
subsets, and functions.
Theorem 3.6. Let X be a first countable space. Then X is Hausdor↵ if and only if
the limits of convergent sequences are unique.
3.3 Nets and three theorems about them
33
Proof. Suppose that X is first countable. If X is not Hausdor↵, 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 2 Un \ Vn , ;. The sequence {x n } has a subsequence that converges to x and to
y.
Theorem 3.7. 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).
Proof. Exercise.
Theorem 3.8. Let X be a first countable space and let A ✓ X. A point x 2 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 3.5. A directed set is a pair (S, ) where S is a set and  is a relation on
S satisfying:
• for all s 2 S, s  s
• for all s,t,u 2 S, s  t and t  u imply s  u
• for all s,t 2 S, there exists a u 2 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 directe set be a directed poset—it is not necessary to assume that the relation be anti-symmetric as is done in [16]. 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 i↵ 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 2 X. Then we can define a
directed set by (X, ) where y  z i↵ 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.
34
3 Limits
Example 3.12. Let X be a topological space and let x 2 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 i↵ 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 3.6. 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 2 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 2 U there exists a t 2 S so that for all t  s,
x s 2 U.
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 [14, 15] (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
i=1 , x 2[ x i 1, x i ]
-
x i 1 ) and L P =
n
X
i=1
inf
x 2[ 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 }
Rb
converges 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 2 X. For every neighborhood
U of x, choose a point xU 2 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 3.9. A space is Hausdor↵ if and only if limits of convergent nets is unique.
Proof. Exercise.
Theorem 3.10. 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 3.11. A point x 2 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 2 A.
Conversely, if x 2 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 Categorical limits
35
3.4 Categorical limits
We’ve discussed limits of sequences and nets. We can also talk about categorical
limits and colimits, of which all the constructions so far, as well as the closure A
and interior A of a set A are just particular examples. The reader is invited to skip
ahead and return to this section later.
First, it’s helpful to 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
! •
!
•
!
•
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:
-
f
X
is a diagram
!Y
!
!
•
+/
/
! C///
//
g
h
!
,
! •
!
*. •
.
a functor ...
..
Z
Definition 3.7. 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.
By viewing an object A of C as a constant functor D ! C, 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
36
3 Limits
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 so that fit together as pictured below.
f
! FY
!
fY
!
fX
!
FX
A
g
h
fZ
!
!
FZ
Now, we can define the limit of a D-shaped diagram F : D ! C.
Definition 3.8. 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.
Ff
! FY
!
fY
fX
!
!
!
FX
!
A
gX
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 3.9. 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.
3.4 Categorical limits
37
Ff
fX
!
A
!
FX
! FY
fY
gX
gY
h
!
!
!
B
Examples help to understand limits and colimits. Work through these:
Example 3.16. The limit of the empty diagram is a terminal object and the colimit
of an empty diagram is an initial object.
Example 3.17. 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.
Example 3.18. The colimit of the diagram
g
Z
! X
!
f
Y
is called the pushout of X and Y along the morphisms f and g. For an example, for
any two sets, we have the diagram
- !
A\ B -
i
! A
i
B
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:
i
B
i
! A
!
- !
A\ B -
! A[ B
!
!
! S
Example 3.19. 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
38
3 Limits
`
X t Z Y = X Y/⇠ where x ⇠ y , f (x) = g(y). This pushout satisfies the
universal property described by this diagram:
f
Z
! X
g
Y
!
!
iX
iY
! X tZ Y
!
!
!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 Z 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 Z to a space W are specified by maps a : X ! W and
b : Y ! W with f a = gb.
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.
Sn
f
1
! X
iX
Dn
iY
!
!
i
! X tS n
1
Dn
Example 3.20. 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, the pullback
is realized by X ⇥ Z Y = {(x, y) : f (x) = g(y)}.
Example 3.21. The limit of the diagram
X
f
!Y
g!
is called the equalizer of f and g. In Set, the equalizer can be realized as the set
E = eq( f , g) = {x 2 X : f (x) = g(x)} with the inclusion E ! X. Here’s the
picture of its universal property:
3.4 Categorical limits
39
!
E
f
! X
!
!Y
g!
S
In groups, the equalizer of f : G ! H and 0 : G ! H is the kernel of f .
Example 3.22. The colimit of the diagram
X
f
!Y
g!
is called the coequalizer of f and g. In Set, the coequalizer is realized as the quotient
of the set Y obtained by identifying f (x) and g(x) for each x 2 X.
Example 3.23. When the diagram category D is a poset, limits are called inverse
limits or projective limits and colimits are called directed limits or inductive limits.
This terminology is confusing, but it’s out there so you might as well know about it.
Theorem 3.12. If a category C has products and equalizers then it has all limits. If
it has all colimits and coequalizers then it has all colimits.
Proof. Exercise.
Exercises
3.1. Prove that R with the co-countable topology (sets with countable compliment
are open) is a non-Hausdor↵ space in which convergent sequences have unique limits.
3.2. Show that the directed sets defined in the 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 3.9.
3.6. Prove theorem 3.10.
3.7. Here are two variations of Hausdor↵. 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
Hausdor↵ implies KC implies U S, but that the implications are strict [20].
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.)
40
3 Limits
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. 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
Tychono↵’s Theorem (optional)
The Axiom of Choice is obviously true, the well-ordering theorem is obviously false; and
who can tell about Zorn’s Lemma?
— Jerry Bona
The goal of this chapter is to prove
Tychono↵’s Theorem. The product of compact spaces is compact.
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 [16], 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 [11] states
Tychono↵’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, and it’s
perfectly accurate to say “must” use the axiom of choice since in 1950, Kelley
proved that Tychono↵’s theorem implies the axiom of choice [10]. It’s worth noting
that there’s a two line proof of Tychono↵’s theorem in [21] using “ultrafilters,” due
originally to Cartan [2], but it would be rather more than two lines to develop that
machinery for just this theorem.
4.1 Preliminaries from set theory
In a formal proof of Tychono↵’s theorem, there’s no way to avoid some set-theoretic
details.
41
42
4 Tychono↵’s Theorem (optional)
4.1.1 Axiom of choice and Zorn’s lemma
For convenience, we restate the axiom of choice.
The Axiom of Choice. Given any collection of nonempty sets {X↵ }↵ 2 A , the product
Q
↵ 2 A X ↵ 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 ! [↵ 2 A X↵ with f (↵) 2 X↵ for each
↵ 2 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 Tychono↵’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 i↵ for every
a, b 2 C either a  b or b  a. An element b 2 P is called an upper bound for a
subset A ⇢ P provided a  b for all a 2 A. A subset A ✓ P has an upper bound i↵
there exists b 2 P that is an upper bound for A. We say that m is a maximal element
of a partially ordered set P i↵ there exists no element a 2 P with m  a and a , m.
One can write a < b if a  b and a , b. 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 4.1. 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 2 P, define a set Ea := {b 2
P : a < b}. If Ea = ; for any a, then a is a maximal element of P and we’re done.
If, however, Ea , ; for any a, then the axiom of choice says there is a function
f : P ! [a 2P Ea with f (a) 2 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 {X↵ }↵ 2 A be a
collection of nonempty sets. Define a partial choice function to be a function
4.2 Nets and compactness
43
f : I ! [↵ 2I X↵ 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 i↵ 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 Hausdor↵ 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. It’s worth pointing out the inconsistent
use of terminology here: equivalent statements might be axioms, lemmas, or even
principles.
4.2 Nets and compactness
We introduce the notion of a cluster point of a nets.
Definition 4.1. Let X be a space and let {x ↵ }↵ 2⇤ be a net. We say that a point x 2 X
is a cluster point (or accumulation point) of the net {x ↵ }↵ 2⇤ if and only if for every
open set U containing x and every ↵ 2 ⇤, there exists
2 ⇤ with ↵ 
and
x 2 U.
Example 4.1. Consider the net {x n }n 2N in R given by
1
8
>
n
>
>
>
>
>1
xn = <
>
>
n
>
>
>
>
:2
1
n
if n = 1, 5, 9, 13, . . .
if n = 2, 6, 10, 14, . . .
if n = 3, 7, 11, 15, . . .
if n = 4, 8, 12, 16, . . .
The first few terms of {x n } are
1
1 5
1 9
1 13
1, , 3, 2, , , 7, 2, , , 11, 2, , , 15, 2 . . .
2
5 6
9 10
13 14
44
4 Tychono↵’s Theorem (optional)
Then 0, 1, 2 are all cluster points of this net. Note that the limit points of the range
of this net are 0 and 1.
Theorem 4.2. 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. (b) ) (c). Let {x ↵ }↵ 2⇤ be a net. For each ↵ 2 ⇤ define
F↵ = {x : ↵ 
} and E↵ = F↵ .
Since ⇤ is a directed set, if ↵, ↵ 0 2 ⇤ there exists 2 ⇤ with ↵  and ↵ 0  .
Then x 2 F↵ \ F↵0 ✓ E↵ \ E↵0 . It follows that {E↵ } is a collection of closed sets
with the FIP. Therefore, there is an element x 2 \E↵ . We claim that x is a cluster
point of the net {x ↵ }. To see this,let U be an open set containing x and let ↵ 2 ⇤.
Since x 2 E↵ = F↵ , there is a point y 2 F↵ \ U. By definition of F↵ , y = x for
some ↵  , as needed.
(c) ) (a). Let {U↵ }↵ 2⇤ be a collection of open sets for which no finite subset
covers X. We’ll prove that {U↵ } is not a cover of X. Let D be the set of finite subsets
of ⇤ directed by inclusion: F  G , G ✓ F. For each F 2 D, [↵ 2F U↵ is not
a cover of X, therefore there exists a point x F 2 X \ ([↵ 2F U↵ ). 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 < [↵ 2⇤U↵ . To see this, fix ↵ 2 ⇤. Then the singleton set {↵} 2 D,
therefore, for any open set U containing x, there is a finite set F with {↵} ✓ F and
a point x F 2 U. This says that x F 2 X \ ([ 2F U ) ✓ X \ U↵ . This proves that
x 2 X \ U↵ = X \ U↵ . Since this is true for every ↵, x < [↵ 2⇤U↵ .
Definition 4.2. A subnet of a net f : S ! X is the composition f
where :
T ! S is an increasing cofinal function from a directed set T. Increasing means that
(a)  (b) whenever a  b and cofinal means that for every s 2 S, there exists an
t 2 T with s  (t).
Corollary 4.1. A space is compact if and only if every net has a convergent subnet.
One might ask whether first countable spaces with the property that every sequence has a convergent subsequence are compact. The answer is no, not in general. The reader who is interested in looking for examples may want to know that
the property that every sequence has a convergent subsequence is called sequentially
compact.
4.3 A proof of Tychono↵’s Theorem
Here, we give the proof due to Cherno↵ in 1992 [3].
4.4 Tychono↵’s theorem implies the axiom of choice
45
Proof (Proof of Tychono↵’s theorem). Let {X↵ }↵ 2 A be a family of compact spaces,
Q
let X = ↵ 2 A X↵ and let { f d }d 2D be a net in X. As in the proof of Theorem 4.1.1,
Q
we define an element of ↵ 2I X↵ 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 ↵ 2 A be one index and set I = {↵}, then { f d (↵)} is a net in X↵ . Since
X↵ is compact, there is a cluster point p 2 X↵ of the net { f d (↵)}. Then for I = {↵},
the partial function f : I ! X defined by f (↵) = 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 = A then we are done. If
the domain of g , A, choose an index ↵ 2 A \ I. Since g is a cluster point of the
net { f d | I }d 2D , there’s a subnet { f (e) | I }e 2E converging to g. Since X↵ is compact,
there’s a cluster point p 2 X↵ of the net { f (e) (↵)}e 2E . Then h defined by
8
>p
h( ) = <
> g( )
:
if
if
=↵
2I
is a partial cluster point with domain I [ {↵} extending g. This contradicts the maximality of g. Therefore, the domain of g is all of A and the proof is complete.
4.4 Tychono↵’s theorem implies the axiom of choice
We used Zorn’s lemma to prove the Tychono↵ theorem. In 1950, Kelley proved
that the Tychono↵ theorem is equivalent to the axiom of choice [10]. In order to
prove that the Tychono↵ 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 4.3. The Tychono↵ theorem , the axiom of choice.
Proof. We used Zorn’s lemma to prove Tychono↵’s theorem, which shows that Tychono↵’s theorem is implied by the axiom of choice.
To prove that Tychono↵’s theorem implies the axiom of choice, let {X↵ }↵ 2 A be a
collection of nonempty sets. We need to make a bunch of compact spaces so we can
apply the Tychono↵ theorem. First, let Y↵ = X↵ [ {1↵ }; we add a new element to
X↵ called “1↵ ” We make each Y↵ into a topological space by defining the topology
to be {;, {1↵ }, X↵ ,Y↵ }. Note that Y↵ is compact—there are only finitely many open
46
4 Tychono↵’s Theorem (optional)
Q
sets so every open cover is finite. So, by Tychono↵’s theorem, Y := ↵ 2 A Y↵ is
compact.
Now consider a collection of open sets {U } 2 A of Y where U is the basic open
set in Y obtained by taking the product of all Y↵ ’s for ↵ , and in the factor
put the open set {1 }. Notice that any finite subcollection {U 1 , . . . ,U n } cannot
n U : choose a partial
cover Y for the function f defined as follows is not in [i=
i
1
Qn
function f 2 i=1 X i which is possible without the axiom of choice since the
product is finite. Then extend f to a function f 2 Y be setting f (↵) = 1↵ for all
↵ , 1 , . . . , n , which is possible since we’re not making any choices.
Therefore, the collection {U } cannot cover Y . So, there is a function f 2 Y not
in the [↵ 2 AU↵ . This says that for no ↵ 2 A does f (↵) = 1↵ . Therefore, f (↵) 2 X↵
for each ↵, which is a desired choice function.
4.5 Exercises
4.1. 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 3.6 whose corners have rational
coordinates.
4.2. Prove that x is a cluster point of a net if and only if there exists a subnet converging to x.
4.3. What’s wrong with the following easy “proof” of Tychono↵’s theorem?
Q
Proof (Wrong proof). Let {X↵ }↵ 2 A be a family of compact spaces, let X = ↵ 2 A
and let { f d }d 2D be a net in X. For each ↵ 2 A, we have a net { f d (↵)} in X↵ . Since
X↵ is compact, the net { f d (↵)} has a cluster point p↵ 2 X↵ . Then, the function
f 2 X defined by f (↵) = p↵ is a cluster point of the net { f d }, proving that X is
compact.
Try giving a clear illustration of why this proof is wrong by using the product of just
two compact spaces.