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Topic: Nonempty intersection
Replies: 5   Last Post: Jul 21, 2011 11:32 PM

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William Elliot

Posts: 1,948
Registered: 5/30/08
Re: Nonempty intersection
Posted: Jul 21, 2011 11:32 PM
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On Thu, 21 Jul 2011, Kaba wrote:
> Rotwang wrote:
>> On 21/07/2011 17:37, Kaba wrote:
>>>
>>> Here's an exercise from the book "Introduction to Topological
>>> Manifolds", page 97:
>>>
>>> "Let X be a compact space, and suppose {F_n} is a countable collection
>>> of nonempty closed subsets X that are nested, which means that
>>> F_n superset F_{n + 1} for each n. Show that intersection {F_n} is
>>> nonempty."
>>>
>>> Hints?

>>
>> Suppose that the intersection is empty. Note that X is the complement of
>> the empty set, and apply De Morgan's law to conclude that the union of
>> the complements of the F_n's is X. Since X is compact...

>
> ... there is a finite set of F_n's whose complements cover X. Using De
> Morgan again, the intersection of those F_n's is empty. Because the
> F_n's are nested, some of the F_n's must be empty, which is a
> contradiction. Therefore, the countable intersection is nonempty.
>

Generalization. If S is a compact space, C a collection of closed
subsets, U an open subset of S and the big intersection of C a subset
of U, then there's some K1,.. K_n in C with K1 /\../\ K_n subset U.




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