Exercise · ZW-CLOSURE-OF-A-SET-007 · ★★☆ Intermediate
Let \(A\) be a subset of a topological space \(X\). Which statement is true?
Choose one option.
Show answer
- C. \(\overline{A}=A\) if and only if \(A\) is closed
Explanation
The closure is the smallest closed set containing \(A\); it equals \(A\) exactly when \(A\) is already closed. Always \(A\subseteq\overline{A}\), not the other way around.
Review the concept: Closure of a set
The closure \(\overline{A}\) of a subset \(A\) of a topological space \((X,\mathcal{T})\) is the smallest closed set \(F\) of \(X\) containing \(A\).
\[\overline{A}=\bigcap\{F\ \text{closed}:\ A\subseteq F\}\]
More practice problems: closure of a set
- ZW-CLOSURE-OF-A-SET-001 In X=\1,2,3\ with T=\∅,\1\,\1,2\,\1,2,3\\, what is the closure of \3\?
- ZW-CLOSURE-OF-A-SET-002 In X=\1,2,3\ with T=\∅,\1\,\1,2\,\1,2,3\\, what is the closure of \2\?
- ZW-CLOSURE-OF-A-SET-003 In X=\a,b,c,d\ with T=\∅,\a\,\b\,\a,b\,\a,b,c\,\a,b,c,d\\, what is the closure of \a,b\?
- ZW-CLOSURE-OF-A-SET-004 In X=\1,2,3,4\ with T=\∅,\1\,\3\,\1,3\,\1,2,3\,\1,3,4\,\1,2,3,4\\, what is the closure of \1,2\?
- ZW-CLOSURE-OF-A-SET-005 In ℝ with the usual topology, what is the closure of (0,1)?
- ZW-CLOSURE-OF-A-SET-006 In ℝ with the usual topology, what is the closure of ℚ?
- ZW-CLOSURE-OF-A-SET-008 In usual ℝ, let A=(0,1) and B=(1,2). What is overlineA∩overlineB?
- ZW-CLOSURE-OF-A-SET-009 In a space with the trivial topology \∅,X\, what is the closure of a nonempty subset A?