Lecture Notes Topology (2301631) Phichet Chaoha Department of ...

26 Topology (2301631)
7. Quotient Topology
Definition 1.115. Let X and Y be spaces. A surjective map p : X → Y is
called a quotient map (or an identification) if for each V ⊆ Y , V is open in Y iff
p −1 (V ) is open in X.
Remark 1.116. Every quotient map is always continuous, but not vice versa.
Note also that, the condition in the definition above is equivalent to ”C is closed in
Y iff p −1 (C) is closed in X”, but NOT to ”U is open in X iff p(U) is open in Y ”.
Definition 1.117. Let X and Y be spaces. A map f : X → Y is said to be
open if for each open subset U of X, the set f(U) is open in Y . A closed map is
defined in the similar manner.
Theorem 1.118. Every continuous surjective map that is either open or closed
is a quotient map.
Proof. Let f : X → Y be a continuous surjective map that is open, and
V ⊆ Y . By the continuity of f, if V is open in Y , then f −1 (V ) is open in X.
Conversely, assume that f −1 (V ) is open in X. Since f is open and surjective,
then V = f(f −1 (V )) is open in Y . The case where f is closed can be proved
similarly.
□
Exercise 1.119. Find (and justify your answers) :
(1) a surjective continuous map that is not a quotient map.
(2) a quotient map that is neither open nor closed.
Exercise 1.120. Let {X α } α∈Λ be a collection of spaces. Show that for each
β ∈ Λ, the projection map π β : ∏ α∈Λ X α → X β is always open, but it may not be
closed.
Definition 1.121. Let (X, τ X ) be a space, A a set and p : X → A a surjective
map. The quotient topology τ p on A induced by p is the unique topology on A
which makes p a quotient map; i.e.,
τ p = {U ⊆ A | p −1 (U) ∈ τ X }.
Example 1.122. Consider the surjective map p : R → {a, b, c} defined by
⎧
⎪⎨ a ; x < 0
p(x) = b ; x = 0
⎪⎩
c ; x > 0.
Then the quotient topology on {a, b, c} induced by p is {∅, {a, b, c}, {a}, {c}, {a, c}}.
Definition 1.123. Let X be a space, A a partition of X into disjoint subsets
and p : X → A the surjective map that sends each point in X to the element of
A containing it. The set A together with the quotient topology induced by p is
usually called a quotient space (or an identification space) of X. Moreover, if ∼ is
the equivalence relation on X induced by A, we will usually denote the quotient
space A by X/ ∼.
Remark 1.124. Since a subset V of the quotient space X/ ∼ is just a collection
of equivalence classes, then V is open in X/ ∼ iff ∪ V is open in X.
Example 1.125. Let I = [0, 1], and I 2 = I × I.