Exercise · ZW-COMPACT-SPACE-005 · ★☆☆ Basic
Which subset of \(\mathbb{R}\) is compact?
Choose one option.
Show answer
- B. \(\{0\}\cup\{1/n : n\ge 1\}\)
Explanation
The set \(\{0\}\cup\{1/n\}\) is closed (it contains its only limit point, \(0\)) and bounded. The others are not closed in \(\mathbb{R}\), hence not compact.
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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- ZW-COMPACT-SPACE-009 Let K_1,K_2 be compact subsets of a space X. Which statement is true?