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Topic: intersection of embedded closed sets (Read 1983 times) |
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MonicaMath
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intersection of embedded closed sets
« on: Sep 17th, 2009, 11:55am » |
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Hi,
I need to prove that?
if {A_k}, k=1,..., infinity, is a collection of nonempty embedded closed sets of real numbers in decreasing order with A_j is bounded for one j, then :
the intersection is nonempty ??
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Eigenray
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Re: intersection of embedded closed sets
« Reply #1 on: Sep 17th, 2009, 6:12pm » |
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Are you familiar with the open cover definition of compactness?
If the intersection were empty, we would have
Aj =  k>j Uk,
where Uk = Aj \ Ak is open in Aj. Since Aj is compact, this open cover has a finite subcover. But the Uk are nested increasing, so we must have Aj = Uk for some k, meaning Ak is empty, a contradiction.
There is a more general version here.
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