\( \newcommand{\cat}[1] {\mathrm{#1}} \newcommand{\catobj}[1] {\operatorname{Obj}(\mathrm{#1})} \newcommand{\cathom}[1] {\operatorname{Hom}_{\cat{#1}}} \)

Compactness. Subspaces

How do compactness and subspaces interact?

1. A subspace of a compact space [is compact/is not necessarily compact?].
2. Every closed subspace of a compact space [is compact/is not necessarily compact?].
3. Every compact subspace of a compact space [is closed/is not necessarily closed?].
4. Every compact subspace of a Hausdorff space [is closed/is not necessarily closed?].
5. A subspace of a compact Hausdorff space is [ something iff something].