Exercise · ZW-OPEN-COVER-008 · ★★☆ Intermediate
In \(X=\{1,2,3\}\) with open sets \(\varnothing,\{1\},\{1,2\},X\), how many subfamilies of \(\{\{1\},\{1,2\},X\}\) cover \(X\)?
Choose one option.
Show answer
- A. 4
Explanation
Every cover must contain \(X\), and the other two open sets may or may not be present: \(2^2=4\).
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-009 Does the cover \(1/n,1): n≥2\ of the interval (0,1) admit a finite subcover?