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Problem Of The Week # 321 - Aug 31, 2018

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Euge

MHB Global Moderator
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Jun 20, 2014
1,925
Here is this week's POTW:

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Prove that in a compact topological space, any decreasing sequence of nonempty closed sets has non-empty intersection.

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Remember to read the POTW submission guidelines to find out how to submit your answers!
 
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Euge

MHB Global Moderator
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Jun 20, 2014
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Congratulations to castor28 for his correct solution, which is as follows:


Let $E$ be a compact topological space and $\{A_i\}$ a decreasing sequence of non-empty closed sets; $A_i^c$ is therefore an increasing sequence of open proper subsets of $E$.

Assume that $\bigcap A_i= \emptyset$. $\bigcup A_i^c=E$, and the $A_i$ constitute an open cover of $E$. As $E$ is compact, $\{A_i^c\}$ contains a finite sub-cover $\{A_{i_1}^c,\ldots,A_{i_n}^c\}$ whose union is $E$. As the $A_i^c$ constitute an increasing sequence, we have $ A_{i_n}^c=E$, and this contradicts the fact that the $A_i^c$ are proper subsets (because the $A_i$ are non-empty sets).
 
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