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alence relation on
Z
and the space we have in mind is just the associated quotient
space. To make this precise we need to deal with some technical details. We have to
define a disjoint union operation that will handle the case where
X
and
Y
are not
disjoint.
Definition.
The
disjoint union
of two topological spaces
X
and
Y
, denoted by
X
+
Y
,
is defined to be the topological space consisting of the set
XYX0Y1
+= ¥»¥
and the topology whose open sets are
{
}
UVU
¥» ¥
0
1
is open in
XV
and
is open in
Y
.
The spaces
X
and
Y
will always be considered as subspaces of
X
+
Y
under the natural
identifications of
x
X
with (
x
,0) and
y
Y
with (
y
,1).
It is easy to check that the open sets of
X
+
Y
do form a topology, so that we do
have a topological space and subspaces
X
and
Y
.
Returning to our map f :
B
Æ
X
, we can use it to define an equivalence relation ~
f
on
X
+
Y
.
Definition.
Y
and let f :
B
Æ
X
be a continuous map. Let ~
f
be the equiva-
lence relation on
X
+
Y
induced by the pairs (
b
,f(
b
)),
b
Let
B
B
. Define
(
)
XYXY
»=+
~.
f
We say that
X
»
f
Y
is obtained from
X
by
attaching
Y
by f
and call the map f the
attaching map
.
5.4.6. Theorem.
Let p :
X
+
Y
Æ
X
»
f
Y
be the quotient map.
(1) p(
Y
-
B
) is open in
X
»
f
Y
and p maps
Y
-
B
homeomorphically onto p(
Y
-
B
).
(2) p(
X
) is closed in
X
»
f
Y
and p maps
X
homeomorphically onto p(
X
).
Proof.
See [Eise74].
5.4.7. Example.
Let
D
be the unit disk
D
2
,
H
the rectangle [-1,1] ¥ [-1,1], and
B
the left and right ends -1 ¥ [-1,1] » 1 ¥ [-1,1] of
H
. Define f :
B
Æ
D
by
p
p
p
p
Ê
Ë
ˆ
¯
()
=
Ê
Ë
ˆ
¯
(
)
=-
f
1
,
t
cos
6
t
, sin
6
t
and
f
1
,
t
cos
6
t
, sin
6
t
.
Then
D
»
f
H
is topologically (homeomorphic to) a disk with a handle. See Figure 5.9.
Sometimes one has a collection of subsets of a set that already have a topology
and one wants to extend these topologies to a topology of the whole set. We shall see
examples of this in the next chapter.
