Download SAM III General Topology

SAM III
General
Topology
Lecture 2
Contents
Continuous
maps
Continuity
Categories
Top
SAM III
General Topology
Topologies
The ordered set
of topologies
Topologies of
constructed
spaces
Lecture 2
SAM — Seminar in Abstract Mathematics [Version 20130218]
is created by Zurab Janelidze at Mathematics Division, Stellenbosch Univeristy
SAM III
General
Topology
Lecture 2
Contents
Continuous
maps
Continuity
Categories
Top
1 Continuous maps
Continuity
Categories
Top
Topologies
The ordered set
of topologies
Topologies of
constructed
spaces
2 Topologies
The ordered set of topologies
Topologies of constructed spaces
Continuity
SAM III
General
Topology
Lecture 2
Contents
Continuous
maps
Continuity
Categories
Top
Definition
A continuous map f from a topological space (X , τ ) to a topological space
(Y , σ), displayed as
f : (X , τ ) → (Y , σ),
is a function f : X → Y such that
∀A [(A ∈ σ) ⇒ (f −1 (A) ∈ τ )].
Topologies
The ordered set
of topologies
Topologies of
constructed
spaces
Continuous maps between Alexandroff spaces
Describe continuous maps between Alexandroff spaces.
Continuous maps between linearly ordered sets
Describe continuous maps between topological spaces arising from linearly
ordered sets.
The definition of a category
SAM III
General
Topology
Lecture 2
Definition
A category C consists of
a collection of objects,
Contents
Continuous
maps
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for any two objects X and Y in C, a collection hom(X , Y ) of morphisms
from X to Y ,
for any object X in C a distinguished element 1X ∈ hom(X , X ),
and for any three objects X , Y and Z , a map
Topologies
The ordered set
of topologies
Topologies of
constructed
spaces
hom(X , Y ) × hom(Y , Z )
◦
/ hom(X , Z )
called composition of morphisms.
This structure is subject to the following axioms:
Each 1X is an identity for composition, i.e. for any f ∈ hom(W , X ) and
g ∈ hom(X , Y ) we have 1X ◦ f = f and g ◦ 1X = g .
Composition is associative, i.e. h ◦ (g ◦ f ) = (h ◦ g ) ◦ f for all
f ∈ hom(W , X ), g ∈ hom(X , Y ), and h ∈ hom(Y , Z ).
Topological spaces form a category
SAM III
General
Topology
Lecture 2
Contents
Continuous
maps
Continuity
Categories
Top
Topologies
The ordered set
of topologies
Topologies of
constructed
spaces
The category of topological spaces and continuous maps
Show that the usual set-theoretical composite of two continuous maps is
continuous. Deduce that topological spaces and continuous maps form a
category. This category is denoted by Top.
Isomorphisms
In Top, describe those morphisms f : X → Y for which there is a morphism
g : Y → X such that f ◦ g = 1Y and g ◦ f = 1X . In a category, such f is called
an isomorphism and g is called its inverse. Show that in any category, each
isomorphism f has exactly one inverse g , and moreover, g is itself an isomorphism
whose inverse is f . The inverse g of an isomorphism f is usually denoted by
g = f −1 . In a category, two objects X and Y are called isomorphic if there exists
an isomorphism f : X → Y . Show that the relation “is isomorphic to” is an
equivalence relation. Show that in Top, two objects are isomorphic if and only if
as spaces they are homeomorphic.
The ordered set of topologies on a set
SAM III
General
Topology
Lecture 2
Contents
Continuous
maps
Continuity
Categories
Top
Topologies
The ordered set
of topologies
Topologies of
constructed
spaces
Finer and coarser topologies
For a set X , consider the set T (X ) of all topologies on X . Given two topologies τ
and σ on X , we say that τ is coarser than σ (or that σ is finer than τ ) when
τ ⊆ σ. Verify that (T (X ), ⊆) is an ordered set.
Definition
Consider an ordered set (X , 6), and a subset A ⊆ X . The supremum sup(A) of A
is an element x ∈ X such that a 6 x for all a ∈ A and if for some y ∈ X we again
have a 6 y for all a ∈ A, then x 6 y . The infimum of A, written as inf(A), is
defined dually.
Complete ordered sets
Show that if in an ordered set (X , 6) every subset A 6 X has a supremum, then
every subset also has an infimum. Deduce the converse by duality. An ordered set
having these equivalent properties is said to be a complete ordered set. Show
that for any set X the ordered set (T (X ), ⊆) of all topological spaces on X is
complete.
Topologies of constructed spaces
SAM III
General
Topology
Lecture 2
Contents
Continuous
maps
Continuity
Categories
Top
Topologies
The ordered set
of topologies
Topologies of
constructed
spaces
Topologies on subspaces, quotients, products and sums
Show that a topological space (Y , σ) is a subspace of a topological space (X , τ )
if and only if Y ⊆ X and σ is the infimum of the set T of all topologies on Y
such that the inclusion map
Y → X,
y 7→ y
is continuous. Moreover, show that σ ∈ T . The above map is called the
embedding of a subspace (Y , σ) in (X , σ). Establish similar results for topologies
on quotient spaces, products and sums of spaces (hint: in each case, find the
map(s) which play a similar role as the embedding does above).
Factorization of continuous maps
Show that any continuous map between topological spaces can be presented as a
composite of three continuous maps where the first map is an embedding, the
second map is a continuous bijection, and the third map projects a space onto its
quotient space (mapping an element to its equivalence class).