Exercise · ZW-OPEN-COVER-009 · ★★★ Advanced
Does the cover \(\{(1/n,1): n\ge2\}\) of the interval \((0,1)\) admit a finite subcover?
Choose one option.
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- B. No: every finite subfamily has union \((1/N,1)\) and leaves out points close to \(0\).
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Explanation
It does cover \((0,1)\), but a finite subfamily has as union the largest of its intervals, \((1/N,1)\), which omits the points of \((0,1/N]\).
An open cover of a subset \(A\) of a topological space \(X\) is a family \(\mathcal{U}\) of open sets \(U\) of \(X\) whose union contains \(A\).
\[A\subseteq\bigcup_{U\in\mathcal{U}}U\]
More practice problems: open cover
- ZW-OPEN-COVER-001 Which of these families of open sets is an open cover of ℝ?
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- ZW-OPEN-COVER-003 What is the union of the family \(1/n,1): n≥2\?
- ZW-OPEN-COVER-004 How many nonempty subfamilies of \\1\,\2\,\1,2\\ are covers of X=\1,2\ (discrete topology)?
- ZW-OPEN-COVER-005 A subcover of an open cover U of A is:
- ZW-OPEN-COVER-006 The cover \(-n,n): n≥1\ of ℝ admits a finite subcover for only one of these sets. Which?
- ZW-OPEN-COVER-007 What is the minimum number of open intervals of length 0.2 needed to cover [0,1]?
- ZW-OPEN-COVER-008 In X=\1,2,3\ with open sets ∅,\1\,\1,2\,X, how many subfamilies of \\1\,\1,2\,X\ cover X?