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Math 301/ENAS 513: HW 8 Solution
Triet Le
Remark
1
.
Let
U
be any collection of subsets of a generic set
S
. Then
±
[
U
∈U
U
!
c
=
²
U
∈U
U
c
.
1:
If
K
=
{
K
α
}
is a collection of compact subsets of a metric space (
S,d
) such that
the intersection of every ﬁnite subcollection of
K
is nonempty, then
T
K
α
∈K
K
α
is nonempty.
In particular, if
{
K
n
}
is a sequence of nonempty closed and bounded sets in
R
n
such that
K
n
⊃
K
n
+1
, for
n
≥
1. Then
∩
∞
n
=1
K
n
is not empty.
Solution
1
.
Suppose that
T
K
α
∈K
K
α
=
∅
, then there exists a compact set
K
∈ K
such that
for any
x
∈
K
,
x /
∈
T
K
α
6
=
K
K
α
. In other words,
K
T
³
T
K
α
6
=
K
K
α
´
=
∅
. This implies
K
⊂
S
K
α
6
=
K
K
α
. Hence
U
=
{
K
c
α
:
K
α
6
=
K
}
is an open cover for
K
. Since
K
is compact,
there exists a ﬁnite open subcover,
{
K
α
i
:
i
= 1
,...,n
}
, such that
K
⊂
n
[
i
=1
K
c
α
i
, K
α
i
∈ K
.
By the remark, we kave
∅
=
K
²
±
n
[
i
=1
K
c
α
i
!
c
=
K
²
±
n
²
i
=1
K
c
α
i
!
.
This contradicts the assumption that every ﬁnite collection of
K
is nonempty.
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 Fall '07
 TrietLe
 Math, Sets

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