Does Every Open Set Contain a Compact Set?

[AxiomOfChoice]

Is it true that every open set contains a compact set?

 

[g_edgar]

Well, the empty set is open. And yes, even the empty set contains a compact set.

 

[HallsofIvy]

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.

 

[g_edgar]

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.