Graphics Reference
In-Depth Information
Figure 4.3.
Proving that compact sets are closed.
X
V
x
x
U
x
p
collection {
V
x
} is an open cover of
X
. Since
X
is compact, there is a finite subcover
{
V
x
i
}
1£i£k
. It follows that
k
I
U
=
x
i
i
=
1
is an open neighborhood of
p
contained in
R
n
-
X
and so
R
n
-
X
is open.
Next, we show that
X
is bounded. For every
x
Œ
X
, the ball
B
x
=
B
n
(|
x
|+1)
contains
x
in its interior. Let {
B
x
i
}
1£i£k
be a finite subcover of the open cover {
B
x
}.
If r = max
1£i£k
{|
x
i
|+1}, then
X
Õ
B
n
(r).
The Heine-Borel-Lebesque theorem is one of the fundamental theorems in analy-
sis because it abstracts an essential property of sets that is needed to make many
results dealing with continuous functions valid. Although the “correct” definition of
compactness is in terms of open covers of sets, the theorem is often used to justify
defining a set to be compact if it is closed and bounded. This definition is certainly
easier to understand if one is not very familiar with analysis. Since the concepts are
equivalent, it does not matter much from a practical point of view. At any rate, it
follows that all closed intervals [a,b], all disks
D
n
(
p
,r), and all spheres
S
n
are compact.
The definitions above were all relative to a fixed Euclidean space
R
n
. For example,
if we were to say that a set is open, then we really are, more accurately, saying that
it is open
in R
n
. It is important to have a version of these definitions that is relative
to other spaces besides
R
n
. Specifically, we want to be able to talk about sets that are
open
in X
with respect to any other given space
X
.
Definition.
A subset
A
of a set
X
in
R
n
is said to be
open in
X
, or
relatively open
, if
there is an open set
OinR
n
with the property that
A
=
O
«
X
.
A
is said to be
closed
in
X
, or
relatively closed
, if
X
-
A
is open in
X
.
For example, the subset
q
p
Ó
˛
(
)
A
=
cos ,siqq
0
<<
2
of the plane is
not
open in the plane, but it
is
open in the unit circle
S
1
because
A
is
the intersection of
S
1
with the open set
