Exercise · ZW-CLOSED-SET-007 · ★★☆ Intermediate
Let \((X,\mathcal{T})\) be a topological space. Which statement is true?
Choose one option.
Show answer
- A. The intersection of any family of closed sets is closed
Explanation
By De Morgan's laws, the complement of an intersection of closed sets is the union of their complements, which are open, so it is open. Arbitrary unions fail: in \(\mathbb{R}\), \(\bigcup_n[1/n,1]=(0,1]\) is not 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-006 In any topological space (X,T), 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?