wu :: forums - Print Page


    
      
        wu :: forums
        (http://www.ocf.berkeley.edu/~wwu/cgi-bin/yabb/YaBB.cgi)
      

        general >> wanted >> intersection of embedded closed sets
        
(Message started by: MonicaMath on Sep 17^th, 2009, 11:55am)

Title: intersection of embedded closed sets
Post by MonicaMath on Sep 17^th, 2009, 11:55am

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 ??

Title: Re: intersection of embedded closed sets
Post by Eigenray on Sep 17^th, 2009, 6:12pm

Are you familiar with the open cover definition of compactness?

If the intersection were empty, we would have

A_j = http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/cup.gif_k>j U_k,

where U_k = A_j \ A_k is open in A_j. Since A_j is compact, this open cover has a finite subcover. But the U_k are nested increasing, so we must have A_j = U_k for some k, meaning A_k is empty, a contradiction.

There is a more general version [link=http://planetmath.org/?op=getobj&from=objects&id=4181]here[/link].