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