Let $X$ and $Y$ be topological spaces, $X$ compact, $f : X \to Y$ continuous. Then the preimage of each compact subset of $Y$ is compact.
With the stipulation that $X$ and $Y$ are metric spaces, this is a theorem in Pugh's Real Mathematical Analysis. The proof uses sequential compactness. Is this theorem true in general (i.e. can it be proved with covering compactness alone)?