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Economics 204
Lecture 6–Monday, August 3, 2009
Revised 8/4/09, Revisions indicated by ** and
Sticky Notes
Section 2.8, Compactness
Defnition 1
A collection of sets
U
=
{
U
λ
:
λ
∈
Λ
}
in a metric space (
X, d
)isan
open cover
of
A
if
U
λ
is open for all
λ
∈
Λand
∪
λ
∈
Λ
U
λ
⊇
A
(Λ may be Fnite, countably inFnite, or uncountable.)
Ase
t
A
in a metric space is
compact
if every open cover of
A
contains a Fnite subcover of
A
. In other words, if
{
U
λ
:
λ
∈
Λ
}
is
an open cover of
A
, there exist
n
∈
N
and
λ
1
,
···
,λ
n
∈
Λsuch
that
A
⊆
U
λ
1
∪···∪
U
λ
n
It is important to understand what this defnition does not say.
In particular, it does not say “
A
has a fnite open cover;” note
that every set is contained in
X
,and
X
is open, so every set
has a cover consisting oF exactly one open set. Like the
ε

δ
defnition oF continuity, in which you are given an arbitrary
ε>
0
and are challenged to speciFy an appropriate
δ
,hereyou
are given an arbitrary open cover and challenged to speciFy a
fnite subcover oF the given open cover.
Example:
(0
,
1] is not compact in
E
1
. To see this, let
U
=
⎧
⎪
⎨
⎪
⎩
U
m
=
⎛
⎜
⎝
1
m
,
2
⎞
⎟
⎠
:
m
∈
N
⎫
⎪
⎬
⎪
⎭
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View Full DocumentThen
∪
m
∈
N
U
m
=(0
,
2)
⊃
(0
,
1]
Given any fnite subset
{
U
m
1
,...,U
m
n
}
oF
U
,let
m
=max
{
m
1
,...,m
n
}
Then
∪
n
i
=1
U
m
i
=
U
m
=
⎛
⎜
⎝
1
m
,
2
⎞
⎟
⎠
6⊇
(0
,
1]
so (0
,
1] is not compact.
Note that this argument does not work For [0
,
1]. Given an open
cover
{
U
λ
:
λ
∈
Λ
}
, there must be some
λ
∈
Λ such that 0
∈
U
λ
,
and thereFore
U
λ
⊇
[0
,ε
)Fo
rsom
e
ε>
0, and a fnite number
oF the
U
m
’s we used to cover (0
,
1] would cover the interval (
ε,
1].
This is not a prooF that [0
,
1] is compact, since we need to show
that
every
open cover has a fnite subcover, but it is suggestive,
and we will soon see that [0
,
1] is indeed compact.
Example:
[0
,
∞
) is closed but not compact. To see that [0
,
∞
)is
not compact, let
U
=
{
U
m
=(
−
1
,m
):
m
∈
N
}
Given any fnite subset
{
U
m
1
,...,U
m
n
}
oF
U
,let
m
=max
{
m
1
,...,m
n
}
Then
U
m
1
∪···∪
U
m
n
=(
−
1
,m
)
6⊇
[0
,
∞
)
Theorem 2 (8.14)
Every closed subset
A
of a compact met
ric space
(
X, d
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 Summer '08
 ANDERSON
 Economics

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