Exercise · ZW-OPEN-COVER-006 · ★★☆ Intermediate
The cover \(\{(-n,n): n\ge1\}\) of \(\mathbb{R}\) admits a finite subcover for only one of these sets. Which?
Choose one option.
Show answer
- C. \([-5,5]\)
Explanation
\((-6,6)\) already covers \([-5,5]\). The others are unbounded and any finite subfamily lies inside some \((-N,N)\).
Review the concept: Open cover
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?
- ZW-OPEN-COVER-002 In X=\1,2,3\ with open sets ∅,\1\,\1,2\,X, which family of open sets covers X?
- 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-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?
- ZW-OPEN-COVER-009 Does the cover \(1/n,1): n≥2\ of the interval (0,1) admit a finite subcover?