Exercise · ZW-COMPACT-SPACE-009 · ★★★ Advanced
Let \(K_1,K_2\) be compact subsets of a space \(X\). Which statement is true?
Choose one option.
Show answer
- C. \(K_1\cup K_2\) is compact
Explanation
Given a cover of the union, pick a finite subcover for each \(K_i\) and join them. With infinitely many compacts it fails (\(\mathbb{Z}\) is a union of points) and the intersection may fail without Hausdorff.
Review the concept: Compact space
A topological space \(X\) is compact if every open cover \(\mathcal{U}\) of \(X\) has a finite subcover.
\[X=\bigcup_{U\in\mathcal{U}}U\ \Rightarrow\ X=U_1\cup\dots\cup U_n,\ U_i\in\mathcal{U}\]
More practice problems: compact space
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