Exercise · ZW-COMPACT-SPACE-001 · ★☆☆ Basic
Which of these subsets of \(\mathbb{R}\) is compact?
Choose one option.
Show answer
- C. \([0,1]\)
Explanation
\([0,1]\) is closed and bounded, so by Heine-Borel it is compact. The other three admit open covers with no finite subcover.
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
- ZW-COMPACT-SPACE-002 Is a finite topological space compact?
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- ZW-COMPACT-SPACE-005 Which subset of ℝ is compact?
- ZW-COMPACT-SPACE-006 Let X=\a,b,c,d,e\ with the discrete topology and the cover \a,b\,\c,d\,\e\,\b,c\,\d,e\,\a,e\. What…
- ZW-COMPACT-SPACE-007 Let X be an infinite set with the cofinite topology (the open sets are ∅ and complements of finite…
- ZW-COMPACT-SPACE-008 To prove that A=(0,1) is not compact it suffices to exhibit:
- ZW-COMPACT-SPACE-009 Let K_1,K_2 be compact subsets of a space X. Which statement is true?