Open sets and compact sets

  • #1
Is it true that every open set contains a compact set?
 

Answers and Replies

  • #2
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Well, the empty set is open. And yes, even the empty set contains a compact set.
 
  • #3
HallsofIvy
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Showing that one open set contains a compact set does not prove what the OP wanted!

However, I can finish by showing the "easy" part: If A is not empty, then it contains, say, "a". The singleton set "a", since it is finite, is compact.
 
  • #4
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My point was, that every open set contains the empty open set, which contains the empty compact set. So I was not really doing *one* open set after all.
 

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