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SAM III
General
Topology
Lecture 1
Contents
Ordered sets
Order and
duality
Galois
connections
Topological
spaces
Topology via
open sets
Alexandroff
spaces
Open intervals
Constructing
spaces
Homeomorphism
of spaces
SAM III
General Topology
Lecture 1
SAM — Seminar in Abstract Mathematics [Version 20130202]
is created by Zurab Janelidze at Mathematics Division, Stellenbosch Univeristy
SAM III
General
Topology
Lecture 1
Contents
Ordered sets
1 Ordered sets
Order and duality
Galois connections
Order and
duality
Galois
connections
Topological
spaces
Topology via
open sets
Alexandroff
spaces
Open intervals
Constructing
spaces
Homeomorphism
of spaces
2 Topological spaces
Topology via open sets
Alexandroff spaces
Open intervals
Constructing spaces
Homeomorphism of spaces
Order and duality
SAM III
General
Topology
Lecture 1
Contents
Ordered sets
Order and
duality
Galois
connections
Topological
spaces
Topology via
open sets
Alexandroff
spaces
Open intervals
Constructing
spaces
Homeomorphism
of spaces
Definition
An order 6 on a set X is a binary relation on X (i.e. a subset of X × X ) which is
reflexive, antisymmetric, and transitive. An ordered set is a set equipped with an
order, i.e. it is a pair (X , 6) where X is a set and 6 is an order on X .
Dual of an ordered set
Verify that if (X , 6) is an ordered set, then the pair (X , >) where the relation >
is defined by
x > y ⇔ y 6 x,
is also an ordered set. It is called the dual of the ordered set (X , 6). Show that
the dual of the dual of an ordered set (X , 6) is the same ordered set (X , 6).
Ordered set of subsets
Show that (P(X ), ⊆) is an ordered set.
Galois connections
SAM III
General
Topology
Lecture 1
Contents
Definition
A Galois connection from an ordered set (X , 6) to an ordered set (Y , 6) is a pair
(f , g ) of maps
Ordered sets
Order and
duality
Galois
connections
X
f
g
/Y
which are order-preserving and satisfy
Topological
spaces
Topology via
open sets
Alexandroff
spaces
Open intervals
Constructing
spaces
Homeomorphism
of spaces
o
f (x) 6 y
⇔
x 6 g (y )
for all x ∈ X and y ∈ Y . In a Galois connection (f , g ), the map f is said to be
the left adjoint map, and g is said to be the right adjoint map.
Dual Galois connection
Show that if (f , g ) is a Galois connection from (X , 6) to (Y , 6), then (g , f ) is a
Galois connection from (Y , >) to (X , >).
Galois connections: basic properties
SAM III
General
Topology
Lecture 1
Contents
Ordered sets
Order and
duality
Galois
connections
Topological
spaces
Topology via
open sets
Alexandroff
spaces
Open intervals
Constructing
spaces
Homeomorphism
of spaces
A characterization of Galois connections
Show that a pair (f , g ) of order-preserving maps X o
f
g
/ Y is a Galois
connection from an ordered set (X , 6) to an ordered set (Y , 6) if and only
if x 6 g (f (x)) for all x ∈ X , and f (g (y )) 6 y for all y ∈ Y .
Uniqueness of adjoints
Prove that if (f , g ) and (f , g 0 ) are Galois connections from an ordered set
(X , 6) to an ordered set (Y , 6), then g = g 0 . By duality, deduce a similar
fact for two Galois connections (f , g ) and (f 0 , g ).
Galois connections: examples
SAM III
General
Topology
Lecture 1
Contents
Galois connections induced by functions
Ordered sets
Consider a function f : X → Y from a set X to a set Y . Show that it
induces a Galois connection
Order and
duality
Galois
connections
Topological
spaces
Topology via
open sets
Alexandroff
spaces
Open intervals
Constructing
spaces
Homeomorphism
of spaces
f (−)
P(X ) o
f
−1
/ P(Y )
(−)
from the ordered set (P(X ), ⊆) to the ordered set (P(Y ), ⊆), where
f (A) = {f (a) | a ∈ A} for each A ∈ P(X ).
Topology via open sets
SAM III
General
Topology
Lecture 1
Contents
Ordered sets
Order and
duality
Galois
connections
Topological
spaces
Topology via
open sets
Alexandroff
spaces
Open intervals
Constructing
spaces
Homeomorphism
of spaces
Definition
A topology τ on a set X is a set of subsets of X , τ ⊆ P(X ),
having the following three properties:
{∅, X } ⊆ τ ;
∀A,B [({A, B} ⊆ τ ) ⇒ (A ∩ B ∈ τ )];
S
∀C [(C ⊆ τ ) ⇒ ( C ∈ τ )].
A topological space is a set equipped with a topology, i.e. it is
a pair (X , τ ) where X is a set and τ is a topology on X .
Elements of the topology τ are called open sets of the
topological space, while elements of X are called points of the
topological space.
Alexandroff spaces
SAM III
General
Topology
Lecture 1
Contents
Ordered sets
Order and
duality
Galois
connections
Topological
spaces
Topology via
open sets
Alexandroff
spaces
Open intervals
Constructing
spaces
Homeomorphism
of spaces
Topology is a generalization of order
Verify that for an ordered set (X , 6) the set
τ = {A ∈ P(X ) | ∀x∈A ∀y ∈X [(x 6 y ) ⇒ (y ∈ A)]} is a topology on X . It is called
the Alexandroff topology of (X , 6). Show that two different orders on the same
set give rise to two different Alexandroff topologies.
Alexandroff spaces
An Alexandroff space is a topological space in which arbitrary intersection of
open sets is open. Show that for any fixed set X , there is a bijection between
Alexandroff spaces (X , τ ) and pairs (X , 6) where 6 is a reflexive and transitive
relation. Such pairs are called preorders. The preorder corresponding to an
Alexandroff space is called its specialization preorder. Deduce that Alexandroff
topology is merely the topology in an Alexandroff space corresponding to an
ordered set.
Adjunctions for preorders
Generalize the notion of a Galois connection from ordered sets to preorders,
calling the resulting notion an adjunction between preorders.
Open intervals
SAM III
General
Topology
Lecture 1
Definition
Contents
Ordered sets
Order and
duality
Galois
connections
Topological
spaces
Topology via
open sets
Alexandroff
spaces
Open intervals
Constructing
spaces
Homeomorphism
of spaces
A linearly ordered set is an ordered set (X , 6) such that for any x ∈ X and
y ∈ Y either x 6 y or y 6 x.
Order topology
In an ordered set (X , 6), we write x < y when x 6 y and x 6= y . An open
interval is a subset I ⊆ X satisfying any one of the following three conditions:
I There exists x ∈ X such that I = {i ∈ X | i < x}.
I There exists x ∈ X such that I = {i ∈ X | x < i}.
I There exist x, y ∈ X such that I = {i ∈ X | x < i < y }.
In a linearly ordered set, is the set of arbitrary unions of open intervals a topology?
“Cutting” and “gluing” spaces
SAM III
General
Topology
Lecture 1
Contents
Subspaces
Show that for a topological space (X , τ ) and a subset Y ⊆ X , the set
Ordered sets
Order and
duality
Galois
connections
Topological
spaces
Topology via
open sets
Alexandroff
spaces
Open intervals
Constructing
spaces
Homeomorphism
of spaces
σ = {A ∩ Y | A ∈ τ }
is a topology on Y . The topological space (Y , σ) is called a subspace of (X , τ ).
Quotient spaces
Show that for a topological space (X , τ ) and a partition P of X , the set
σ = {V ⊆ P |
[
V ∈ τ}
is a topology on P, called a quotient topology. The topological space (P, σ) is
called a quotient space.
Adding and multiplying spaces
SAM III
General
Topology
Lecture 1
Contents
Ordered sets
Order and
duality
Galois
connections
Topological
spaces
Topology via
open sets
Alexandroff
spaces
Open intervals
Constructing
spaces
Homeomorphism
of spaces
Product of two spaces
Consider two topological spaces (X , τ ) and (Y , σ). Show that the set
consisting of arbitrary unions of sets of the form A × B, where A ∈ τ and
B ∈ σ, is a topology on X × Y . It is called the Tychonoff topology on the
product X × Y . The corresponding topological space is called the product
of (X , τ ) and (Y , σ).
Sum of two spaces
Consider two topological spaces (X , τ ) and (Y , σ). Show that the set
consisting of sets of disjoint unions A t B where A ∈ τ and B ∈ σ, is a
topology on the disjoint union X t Y . The resulting topological space is
called the sum or coproduct of topological spaces (X , τ ) and (Y , σ).
Product and sum of a family of spaces
Define the product and the sum of an arbitrary family (Xi , τi )i∈I of
topological spaces.
Homeomorphism of spaces
SAM III
General
Topology
Lecture 1
Definition
A topological space (X , τ ) is said to be homeomorphic to a topological
space (Y , σ) when there is a bijection f : X → Y such that for any A ⊆ X ,
Contents
A∈τ
Ordered sets
⇔
f (A) ∈ σ.
Order and
duality
Galois
connections
Topological
spaces
Topology via
open sets
Alexandroff
spaces
Open intervals
Constructing
spaces
Homeomorphism
of spaces
Homeomorphic classes
Show that the relation “homeomorphic” is an equivalence relation on the
class of all topological spaces, i.e. show that it is reflexive, symmetric and
transitive. The corresponding equivalence classes of spaces are called
homeomorphic classes of spaces.
Classification of spaces with at most three points
Describe all homeomorphic classes of topological spaces having at most
three points.