Graphics Reference
In-Depth Information
Definition.
Let
X
be a topological space and let ~ be an equivalence relation on
X
.
The map
p:
XX~
pp
Æ
Æ
[]
which sends an element
p
into its equivalence class [
p
] with respect to ~ is called the
quotient map
.
5.4.1. Lemma.
Let ~ be an equivalence relation on a topological space
X
and let
p:
X
Æ
X
/~ be the quotient map. Define a collection S of subsets of
X
/~ by
{
}
-
1
S
=Õ
BX
/~
p
()
B
is open in
X
.
(5.9)
Then S is a topology on
X
/~.
Proof.
The proof is straightforward. See [Eise74].
Definition.
Let
X
be a topological space and let ~ be an equivalence relation on
X
.
The topology S on
X
/~ defined by equation (5.9), is called the
quotient topology
on
X
/~
and the topological space (
X
/~,S) is called a
quotient space
of
X
.
5.4.2. Lemma.
Let
X
be a topological space and let ~ be an equivalence relation on
X
. The quotient map p :
X
Æ
X
/~ is a continuous map with respect to the quotient
topology on
X
/~.
Proof.
This follows easily from the definition.
5.4.3. Example.
The Moebius strip can be thought of as the quotient space [0,1] ¥
[0,1]/~, where we use the equivalence relation generated by the relation (0,t) ~ (1,1-t), t
[0,1], between the points of the left and right side of the rectangle. See Figure 5.7.
The next theorem lists some basic properties of quotient spaces.
5.4.4. Theorem.
Let ~ be an equivalence relation on a topological space
X
and let
p:
X
Æ
X
/~ be the quotient map. Let
Y
be a topological space.
(1) A map g :
X
/~Æ
Y
is continuous if and only if the composite f = gp:
X
Æ
Y
is continuous.
(2) If f :
X
Æ
Y
is a continuous map that is constant on the equivalence classes of
~, then there is a unique continuous map f* :
X
/~Æ
Y
so that f = f p. The map
f* is called the
induced
(by f )
map
on the quotient space.
o
o
(1,1 - t)
[0,1] ¥ [0,1]
(0,t)
Figure 5.7.
The Moebius strip.
