Exercise · ZW-OPEN-COVER-010 · ★★★ Advanced
\(\mathcal{U}\) covers \(A\) and \(\mathcal{V}\) covers \(B\). Which statement is always true?
Choose one option.
Show answer
- C. \(\mathcal{U}\cup\mathcal{V}\) covers \(A\cup B\).
Explanation
Every point of \(A\cup B\) lies in \(A\) or in \(B\), hence in some member of \(\mathcal{U}\) or \(\mathcal{V}\). The claim about \(\mathcal{U}\cap\mathcal{V}\) fails, for instance, when \(\mathcal{U}\cap\mathcal{V}=\varnothing\) but \(A\cap B\ne\varnothing\).
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-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?