Exercise · ZW-CLOSED-SET-009 · ★★★ Advanced
In \(X=\{1,2,3,4\}\) with \(\mathcal{T}=\{\varnothing,\{1\},\{2\},\{1,2\},\{1,2,3\},X\}\), which of these sets is NOT closed?
Choose one option.
Show answer
- B. \(\{1,2\}\)
Explanation
The complement of \(\{1,2\}\) is \(\{3,4\}\), which is not in \(\mathcal{T}\). By contrast \(\{4\}\), \(\{3,4\}\) and \(\{2,3,4\}\) have complements \(\{1,2,3\}\), \(\{1,2\}\) and \(\{1\}\), all open.
Review the concept: Closed set
A subset \(C\) of a topological space \((X,\mathcal{T})\) is closed when its complement in \(X\) is an open set of the topology \(\mathcal{T}\).
\[C\ \text{closed}\iff X\setminus C\in\mathcal{T}\]
More practice problems: closed set
- ZW-CLOSED-SET-001 In X=\1,2,3\ with T=\∅,\1\,\1,2\,X\, which of these sets is closed?
- ZW-CLOSED-SET-002 How many closed sets does X=\1,2,3\ with T=\∅,\1\,\1,2\,X\ have?
- ZW-CLOSED-SET-003 In the discrete topology on a set X, which subsets are closed?
- ZW-CLOSED-SET-004 In the trivial topology on X=\1,2,3\, which of these sets is closed?
- ZW-CLOSED-SET-005 In ℝ with the usual topology, which of these sets is closed?
- ZW-CLOSED-SET-006 In any topological space (X,T), which statement is true?
- ZW-CLOSED-SET-007 Let (X,T) be a topological space. Which statement is true?
- ZW-CLOSED-SET-008 Let X=\1,2,3,4\ with T=\∅,\1\,\2\,\1,2\,\1,2,3\,X\. How many closed subsets does it have?