Exercise · ZW-COMPACT-SETS-IN-HAUSDORFF-SPACES-001 · ★☆☆ Basic
If \(X\) is Hausdorff and \(K\subseteq X\) is compact, then \(K\) is:
Choose one option.
Show answer
- B. closed in \(X\)
Explanation
This is the central result: in a Hausdorff space compact sets are closed. They need not be open, finite or connected (\([0,1]\cup[2,3]\)).
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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