Graphics Reference
In-Depth Information
Figure 5.11.
The cone and suspen-
sion of a space.
CX
SX
X
X
X
cone on X
suspension
of X
5.5
Compactness
We already pointed out the importance of the compactness property for subsets of R n
in Chapter 4. The generalization to topological spaces is immediate since its defini-
tion was in terms of open sets.
Definition.
X . A collection W of subsets of X is said to be
a cover of A if every element of A belongs to some subset in W, that is,
Let X be a set and let A
U
AU
U
Õ
.
ŒW
If X is a topological space, then we shall call W an open, closed, . . . cover of A if every
set in W is open, closed, ..., in X , respectively. We call W a finite cover if it is a finite
set. A subset of W that covers A is called a subcover of W.
5.5.1. Example.
The collection
{
[
]
}
nn
,
+
1
n
Œ
Z
is a closed cover of R and
1
1
Ó
Ê
Ë
ˆ
¯
˛
nn n
,
1
-
=
23
, ,...
is an open cover of (0,1).
Definition. A topological space X is said to be compact if every open cover of X con-
tains a finite subcover of X .
Let A be a subspace of a topological space X . It is easy to show that A , thought
of as a topological space on its own without reference to X , is compact if and only if
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