Exercise · ZW-COMPACT-SPACE-003 · ★☆☆ Basic
For integers \(n\ge 1\), the intervals \((-n,n)\) cover \(\mathbb{R}\). What is the smallest \(n\) with \([-5,5]\subseteq(-n,n)\)?
Choose one option.
Show answer
- B. \(6\)
Explanation
Since \((-n,n)\) is open, \(\pm5\) fit only when \(n>5\); the least integer is \(6\). So \([-5,5]\) is covered by a single member of the family.
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-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?