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Let A and B be compact subset of R

To show intersection of A and B is compact,

I need to show that for any open cover for intersection has finite subcover.

It is quite straightforward for Union of two compact sets, but how can I start with the intersection casE?

Eric Wofsey
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jessie
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    Given an open cover for the intersection, augment it with the complement of the intersection. Now you have an open cover for the union so you can proceed from there. – John Douma Oct 27 '15 at 01:03

1 Answers1

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Let $A$ and $B$ be compact. Let $U=\{U_\alpha\}$ be an open covering of $A\cap B$. Since we are working in $\mathbb R$, we know that $A$ and $B$ are both closed. Thus, $A\cap B$ is closed, so $U'=U\cup \{\mathbb R\setminus (A\cap B)\}$ is an open cover of $A\cup B$. Now can you use this to construct a finite subcovering of $A\cap B$?

Plutoro
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  • not quite sure. If Uis open cover of union, then U is open cover for A and B. – jessie Oct 27 '15 at 03:31
  • Then A and B has finite subcover and if we take intersection of those two finite subcovers, then it is finite subcover for interseection of A and B? – jessie Oct 27 '15 at 03:32
  • We know that $A\cup B$ is compact, so there is a finite subcover of $A\cup B$. Then we just take $\mathbb R\setminus(A\cap B)$ out of this subcover, we have a finite subcover of $A\cap B$. – Plutoro Oct 27 '15 at 03:40