Exercise · ZW-CLOSED-SET-010 · ★★★ Advanced
Can a subset of a topological space be both open and closed?
Choose one option.
Show answer
- C. Yes: \(\varnothing\) and \(X\) always are, and in the discrete space every subset is
Explanation
«Closed» is not the opposite of «open»: it means the complement is open. \(\varnothing\) and \(X\) satisfy both conditions in every topology, and in the discrete one so does every subset.
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?