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   intersection of embedded closed sets
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   Author  Topic: intersection of embedded closed sets  (Read 1984 times)
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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