1

I want to how that in any metric space,

Intersection of any number of compactsets is compact.

Hosein Rahnama
  • 14,882

1 Answers1

1

Hint (for metric spaces): a compact set is closed; a closed subset of a compact subset is compact; what about intersections of closed sets?

Caveat. “Any number” should be interpreted as “at least one”.

egreg
  • 238,574
  • The intersection of closed set is closed, how can we go further? –  Oct 17 '18 at 15:54
  • @Saeed It is also contained in at least a compact set, isn't it? – egreg Oct 17 '18 at 16:08
  • It is, so it is compact. But this result stems from the fact that we assume the intersection of any number of closed is not empty. To accept this we need that result. Can we proof it using another way? –  Oct 17 '18 at 16:34
  • @Saeed The empty set is obviously compact. – egreg Oct 17 '18 at 17:07