Graphics Reference
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Proof.
The proofs are straightforward. See [Eise74]. A good way to remember this
theorem is in terms of commutative diagrams. Part (1) says that the map g in the
diagram
X
f = g
p
p
X/~
Y
g
is a continuous map if and only it lifts to a continuous map f. Part (2) says that any
continuous map f in the diagram
X
f
p
Y
X/~
f*
that is constant on equivalence classes induces a unique f*.
It is convenient to introduce some notation for a common special case of a quo-
tient space.
Definition.
Let
A
be a nonempty subspace of a topological space
X
. Let ~
A
be the
equivalence relation
=¥»
(
{
)
}
~
AA pppX
,
Œ
.
A
The quotient space
X
/~
A
will be denoted by
X
/
A
and is usually referred to as the space
obtained from
X
by
collapsing
A
to a point
.
The space
X
/
A
is the space we get by identifying all the points of
A
to a single
point
A
, the equivalence class of some
a
A
.
5.4.5. Theorem.
Let
A
be a nonempty subspace of a topological space
X
. If
A
is open
or closed, then the quotient map sends
X
-
A
homeomorphically onto (
X
/
A
) -
A
.
Proof.
Easy.
Let
X
and
Y
be topological spaces. Let
B
be a subspace of
Y
and let f :
B
Æ
X
be a continuous map. We would like to define the space that, intuitively, is obtained
from the disjoint union
Z
of
X
and
Y
where we have “attached” or “glued” each point
b
in
B
to f(
b
) in
X
. See Figure 5.8. The identification of
b
with f(
b
) defines an equiv-
X»
f
Y
Y
Y
X
f(B)
f(b)
X
b
B
Figure 5.8.
Attaching one space to another.
