I am going through Rudin's Principles of Mathematical Analysis in preparation for the masters exam, and I am seeking clarification on a corollary.
Theorem 2.34 states that compact sets in metric spaces are closed. Theorem 2.35 states that closed subsets of compact spaces are compact. As a corollary, Rudin then states that if $L$ is closed and $K$ is compact, then their intersection $L \cap K$ is compact, citing 2.34 and 2.24(b) (intersections of closed sets are closed) to argue that $L \cap K$ is closed, and then using 2.35 to show that $L \cap K$ is compact as a closed subset of a compact set.
Am I correct in believing that this corollary holds for metric spaces, and not in general topological spaces?
Best Answer
In any Hausdorff space compact sets are automatically closed, and so the above argument works as written. It is true that in non-Hausdorff spaces, a compact set need not be closed.
On the other hand, it is true in general that a closed subset of a compact topological space is compact (whether or not the compact space is Hausdorff); this is easily proved directly in terms of the open cover characterization of compact topological spaces.
So if $K$ is compact in an arbitrary topological space (which is just to say that it is a compact topological space when given its induced topology) and $L$ is closed then $K \cap L$ is a closed subset of $K$ in its induced topology, and hence is compact with its induced topology, i.e. is again a compact subset of the ambient topological space (even though it need not be closed).
Summary: The corollary does hold for arbitrary (not necessarily Hausdorff) topological space, but you need to rewrite the proof slightly.