Exercise · ZW-OPEN-COVER-004 · ★☆☆ Basic
How many nonempty subfamilies of \(\{\{1\},\{2\},\{1,2\}\}\) are covers of \(X=\{1,2\}\) (discrete topology)?
Choose one option.
Show answer
- A. 5
Explanation
The 4 subfamilies containing \(\{1,2\}\) cover, and so does \(\{\{1\},\{2\}\}\); 5 in total (checked by listing the 7 nonempty ones).
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-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?
- ZW-OPEN-COVER-009 Does the cover \(1/n,1): n≥2\ of the interval (0,1) admit a finite subcover?