Graphics Reference
In-Depth Information
Figure 5.10.
The wedge of two spaces.
XvY
X
X
x
0
x
0
= y
0
y
0
Y
Y
Unless stated otherwise, whenever one takes a product of topological spaces it will
always be assumed that the product is given the product topology.
Several other constructions that create new spaces from old are handy.
Definition.
A
pointed space
is a pair (
X
,
x
0
), where
X
is a nonempty topological space
and
x
0
X
. The point
x
0
is called the
base point
of the pointed space. The expression
“the pointed space
X
with base point
x
0
” will mean the pointed space (
X
,
x
0
).
Definition.
Let
X
and
Y
be pointed spaces with base points
x
0
and
y
0
, respectively.
Let ~ be the equivalence relation on
X
+
Y
induced by the pair (
x
0
,
y
0
). The
one-point
union
or
wedge
of
X
and
Y
, denoted by
X
⁄
Y
, is defined to be the pointed space that
consists of the quotient space
(
)
~
XY XY
⁄= +
and the point to which
x
0
and
y
0
get identified.
The space
X
⁄
Y
is just the disjoint union of
X
and
Y
where we identify
x
0
and
y
0
. See Figure 5.10.
Definition.
Let
X
be a topological space. Define the
cone on
X
, denoted by C
X
, by
=¥
[]
¥
C
XX
01
,
X
1
.
By identifying
X
with
X
¥ 0 in C
X
, one always considers
X
as contained in C
X
.
See Figure 5.11. Exercise 5.4.2 gives a more concrete description of C
X
.
Definition.
Let
X
be a topological space. Define the
suspension
of
X
, denoted by S
X
,
to be the quotient space
[
]
S
X
=¥-
1,/~,
where ~ is the equivalence relation induced by the relations (
x
,-1) ~ (
x
¢,-1) and (
x
,1)
~ (
x
¢,1) for all
x
,
x
¢
X
. By identifying
X
with
X
¥ 0 in S
X
, one always considers
X
as contained in S
X
.
Again see Figure 5.11. Exercise 5.4.4 gives a more concrete description of S
X
.