Section 7
Econ 204, GSI: Hui Zheng
Key Words
Compactness, Open Cover, Sequentially Compact, Totally Bounded
Section 7.1
A collection of sets
U
=
f
U
:
2
g
in a metric space
(
X;d
)
is an
open cover
of
A
if
U
is open for all
2
and
[
2
U
±
A
.
A set
A
in a metric space is
compact
if every open cover of A
f
U
:
2
g
is an open cover of
A
, there
exist
n
2
N
and
1
n
2
such that
A
²
U
1
[
:::
[
U
n
.
A set
A
in a metric space
(
X;d
)
is
sequentially compact
if
every sequence of elements of
A
contains a convergent subsequence whose limit lies in
A
.
Example 7.1.1
Exhibit an open cover of
(0
;
1)
Solution:
Consider the cover
f
(0
;
1
³
1
=n
)
g
. This covers
(0
;
1)
the form
(0
;
1
³
1
=N
)
, for some
N
. This interval is clearly a proper subset of
(0
;
1)
.
Example 7.1.2
Show that
Q
\
[
0
;
2
]
is not compact.
Solution:
U
n
= (
³
1
;
p
2
³
1
n
)
[
(
p
2 +
1
n
;
3)
. The collection of
sets given by
[
n
2
N
U
n
= (
³
1
;
p
2)
[
(
p
2
;
3)
is an open cover of
Q
\
[
0
;
2
]
. Since
Q
is dense,
N
, there exists a rational number
q
2
Q
\
[
p
2
³
1
N
;
p
2 +
1
N
]
. So
Q
\
[
0
;
2
]
.
Example 7.1.3
Solution:
Let
A
1
;:::;A
n
be compact sets and consider any open cover of
A
1
[´´´[
A
n
. This open cover
must cover each
A
i
individually, and because each
A
i
is compact, there must be a &nite sub
cover of each
A
i
. The union of these
n
A
1
[´´´[
A
n
.
Therefore every open cover of
A
1
[´´´[
A
n
A
1
[´´´[
A
n
is compact.
Example 7.1.4 (Cantor±s Intersection Theorem)
compactness to prove a decreasing sequence of nonempty compact subsets
A
1
µ
A
2
µ ´´´
of a metric space
(
X;d
)
has nonempty intersection.
Solution:
By contradiction. Suppose their intersection in empty:
A
1
\
A
2
\ ´´´
=
±
. Since
A
1
µ
A
2
µ ´´´
and they are nonempty sets,
A
2
\
A
3
\ ´´´
=
±
. Let
U
=
X
n
(
A
2
\
A
3
\ ´´´
) =
X
n
A
2
[
X
n
A
3
[ ´´´
and it is open, so it constructs an open cover for
A
1
. Because
A
1
X
n
A
2
[
X
n
A
3
[ ´´´ [
X
n
A
N
µ
A
1
. Then its
complement
X
n
(
X
n
A
2
[
X
n
A
3
[´´´[
X
n
A
N
) =
A
2
\
A
3
\´´´\
A
N
has no common element
with
A
1
,
A
1
\
A
2
\
A
3
\ ´´´ \
A
N
=
±
. But we know
A
1
\
A
2
\
A
3
\ ´´´ \
A
N
=
A
N
.
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 Summer '08
 ANDERSON
 Topology, Metric space, convergent subsequence, open cover, nite subcover, Ai0

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