What property on a space guarantees that compact sets are closed? What property on a space guarantees that closed sets are compact?
Solution
If a space is Hausdorff, every compact subset is closed.
If a space is compact, every closed subset is compact: if \(F\subseteq X\) is closed and \(\{U_\alpha\}\) is a family of open subsets of \(X\) covering \(F\), then \[ \{U_\alpha\}\cup\{X\setminus F\} \] is an open cover of \(X\); a finite subcover of \(X\) restricts to a finite subcover of \(F\).