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Section 22. The Quotient Topology Note. In this section, we develop a technique that will later allow us a way to visualize certain spaces which cannot be embedded in three dimensions. The idea is to take a piece of a given space and glue parts of the border together. For example,

22 The Quotient Topology 2

Note If C is saturated with respect to p then for some A Y we have p 1 A

Lemma Let X and Y be topological spaces Then p X Y is a quotient map

if and only if p is continuous and maps saturated open sets of X to open sets of Y

Definition The map f X Y is an open map if for each open set U X the

set f U is open in Y The map f X Y is a closed map if for each closed set

A X the set f A is closed in Y

Note If p X Y is continuous and surjective and p is either open or closed

then p is a quotient map However there are quotient maps that are neither open

nor closed see Munkres Exercise 22 3

Example 22 1 Let X 0 1 2 3 be a subspace of R and let Y 0 2 be a

subspace of R Define p X Y as

x if x 0 1

x 1 if x 2 3

Then p is surjective and continuous p is also closed if set A contains limit points

then set f A contains its limit points However p is not open since the open set

0 1 of X is mapped to 0 1 which is not open in Y 0 2

22 The Quotient Topology 3

Example 22 2 Let 1 R R R be projection onto the first coordinate Then

1 is continuous and surjective For any open set O R R O is the countable

union of basis elements of the form U V Then 1 U V U is open in R and

so 1 carries open sets of R R to open sets of R That is 1 is an open map

However 1 is not a closed map The subset C x y xy 1 of R R is

closed in R R but 1 C R 0 which is not closed in R

Definition If X is a space A is a set and p X A is surjective onto map

then there exists exactly one topology T on A relative to which p is a quotient

map This topology is called the quotient topology induced by p

Note The previous definition claims the existence of a topology This topology

is simply the collection of all subsets of set A where p 1 A is open in X This is

in fact a topology since p 1 p 1 A X p 1 J A J p 1 U

where J is an arbitrary set and p 1 ni 1 Ui ni 1 p 1 Ui so each of the parts of

the definition of topology are satisfied The topology is unique since any additional

or less open sets in set A would mean that p is not by definition a quotient map

Example 22 3 Let p be the map of R onto A a b c given by

p x b if x 0

Then the quotient topology on A must be T a b a b a b c

22 The Quotient Topology 4

Note The following idea introduces the technique by which we ll cut and paste

a new space out of a given space

Definition Let X be a topological space and let X be a partition of X into

disjoint subsets whose union is X Let p X X be the surjective onto map

that carries each point of X to the element of X containing it p is called the

projection map from X to X In the quotient topology on X induced by p the

space S under this topology is the quotient space of X

Note Recall that we have a partition of a set if and only if we have an equivalence

relation on the set this is Fraleigh s Theorem 0 22 So Munkres approach in terms

of partitions can be replaced with an approach based on equivalence relations

The idea of the quotient space is that points of the subsets in the partition or

the equivalent points under the equivalence relation are identified with each

other For this reason quotient spaces are sometimes called identifying spaces or

decomposition spaces You will notice the parallel between quotient spaces and

quotient groups in which all elements of a coset are identified

Note In the following three examples the winking smiley faces represent sets

which are equivalent under the equivalence relation induced by the partition X of

X Each of the smiley faces in X is mapped by p to the same place in X

22 The Quotient Topology 5

Example Torus Define an equivalence relation on X R R as x1 y1

x2 y2 if and only if x1 x2 Z and y1 y2 Z We partition X as follows

Then each 1 1 square contains exactly one representative from each equivalence

class Because of the correspondence of boundary points of a 1 1 square the quo

tient space X is homeomorphic to the torus The winking smiley faces represent

sets equivalent under

The image of the torus is from http koc wikia com wiki File Torus jpg

22 The Quotient Topology 6

Example Klein Bottle Define an equivalence relation on X R R as

x1 y1 x2 y2 if and only if

y1 y2 Z and x1 x2 2Z 4 2 0 2 4

y1 y2 Z and x1 x2 Z 2Z 3 1 1 3 5

As in the previous example a 1 1 square contains exactly one representative from

each equivalence class The quotient space X is homeomorphic to the Klein bottle

The image of the Klein bottle is from http www ediblegeography com wp con

tent uploads 2013 03 Klein bottle 460 jpg

22 The Quotient Topology 7

Example Projective Plane Define an equivalence relation on X R R as

x1 y1 x2 y2 if and only if

x1 x2 2Z and y1 y2 2Z or

y1 y2 Z by1c by2c 2Z and x1 x2 Z 2Z or

x1 x2 Z bx1 c bx2 c 2Z and y1 y2 Z 2Z or

y1 y2 Z by1c by2c Z 2Z x1 x2 Z and bx1c bx2 c Z 2Z

In terms of the boundary of the unit square points opposite the center of the

square are identified As in the previous example a 1 1 square contains exactly one

representative from each equivalence class The quotient space X is homeomorphic

to the projective plane An alternate model of the projective plane is given by taking

a closed hemisphere as a surface and identifying points on the boundary which are

opposite the center or antipodal boundary points We ll explore this approach

again in the notes for Munkres Section 60

22 The Quotient Topology 8

Theorem 22 1 Let p X Y be a quotient map Let A be a subspace of X

that is saturated with respect to p Let q A p A be the map obtained by

restricting p to S q p A

1 If A is either open or closed in X then a is a quotient map

2 If p is either an open or a closed map then q is a quotient map

Theorem 22 2 Let p X Y be a quotient map Let Z be a space and let

g X Z be a map that is constant on each set p 1 y for y Y Then g

induces a map f Y Z such that f p g The induced map f is continuous if

and only if g is continuous f is a quotient map if and only if g is a quotient map

22 The Quotient Topology 9

Corollary 22 3 Let g X Z be a surjective continuous map Let X be the

following collection of subsets of X X g 1 z z Z Let X have the

quotient topology

a The map g induces a bijective continuous map f X Z which is a home

omorphism if and only if g is a quotient map

b If Z is Hausdorff so is X

Revised 5 1 2015