Page 107 - DMTH503_TOPOLOGY

Unit 10: The Quotient Topology

10.1 The Quotient Topology Notes

10.1.1 Quotient Map, Open and Closed Map

Quotient Map

Let X and Y be topological spaces; let p : X  Y be a surjective map. The map p is said to be a
–1
quotient map provided a subset U of Y is open in Y if and only if p (U) is open in X.
The condition is stronger than continuity, some mathematicians call it ‘strong continuity’. An
equivalent condition is to require that a subset A of Y be closed in Y if and only if p (A) is closed
–1
in X. Equivalence of the two conditions follow from equation
–1
f (Y – B) = X – f (B).
–1
Open map: A map f : X  Y is said to be an open map if for each open set U of X, the set f(U) is
open in Y.
Closed Map: A map f : X  Y is said to be a closed map if for each closed set A of X, the set f(A)
is closed in Y.

Example 1: Let X be the subspace [0, 1] [2, 3] of and let Y be the subspace [0, 2] of .
The map p : X  Y defined by


 x for x [0,1],
p(x) = 


 x 1 for x [2,3]
is readily seen to be surjective, continuous and closed. Therefore, it is a quotient map. It is not,
however, an open map; the image of the open set [0, 1] of X is not open in Y.

Note If A is the subspace [0, 1] [2, 3] of X, then the map q : A  Y obtained by restricting
p is continuous with surjective but it is not a quotient map. For the set [2, 3] is open in A
and is saturated w.r.t q, but its image is not open in Y.

Example 2: Let  : ×  be projection onto the first coordinate, then  is continuous
1
1
and surjective. Furthermore,  is an open map. For if U × V is a non-empty basis element for
1
× , then  (U × V) = U is open in ; it follows that  carries open sets of × to open sets
1 1
of . However,  is not a closed map. The subset
1
C = {x × y | xy = 1}
of × is closed, but  (C) = – {0}, which is not closed in .
1

Note If A is the subspace of × that is the union of C and the origin {0}, then the map
q : A  obtained by restricting  is continuous and surjective, but it is not a quotient
1
map. For the one-point set {0} is open in A and is saturated with respect to q. But its image
is not open in .

LOVELY PROFESSIONAL UNIVERSITY 101