Exercise · ZW-CLOSED-SET-006 · ★★☆ Intermediate
In any topological space \((X,\mathcal{T})\), which statement is true?
Choose one option.
Show answer
- B. \(\varnothing\) and \(X\) are always closed
Explanation
\(X\setminus X=\varnothing\) and \(X\setminus\varnothing=X\) are open in every topology, so \(X\) and \(\varnothing\) are always closed.
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-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?
- ZW-CLOSED-SET-009 In X=\1,2,3,4\ with T=\∅,\1\,\2\,\1,2\,\1,2,3\,X\, which of these sets is NOT closed?