Theorem. If is a collection of compact subsets of a metric space such that the intersection of every finite subcollection of is nonempty, then
I try and prove theorems my self before reading them (seldom looks pretty), and the one I came up with for this theorem is the following.
Theorem 2.34 says that compact subsets of metric spaces are closed. From this I know that every set in is closed.
Theorem 2.24(b) says that for any collection of closed sets, is closed.
Now comes my question. Since the empty set is both closed and open, can I just say that is nonempty since it is closed? I am pretty sure the answer is no, but it would be nice if someone could explain this a bit.