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 .
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