Exercise · ZW-COMPACT-SETS-IN-HAUSDORFF-SPACES-010 · ★★★ Advanced
Of the \(29\) topologies on \(\{1,2,3\}\), in how many is every compact subset closed?
Choose one option.
Show answer
- C. \(1\)
Explanation
In a finite space every subset is compact, so the condition says every subset is closed: only the discrete topology. This was checked by enumerating all \(29\).
Review the concept: Compact sets in Hausdorff spaces
Every compact subset \(K\) of a Hausdorff space \(X\) is closed in \(X\), and every closed subset of a compact space is compact.
\[K\subseteq X\ \text{compact},\ X\ \text{Hausdorff}\ \Rightarrow\ \overline{K}=K\]
More practice problems: compact sets in hausdorff spaces
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