Exercise · ZW-CLOSURE-OF-A-SET-008 · ★★★ Advanced
In usual \(\mathbb{R}\), let \(A=(0,1)\) and \(B=(1,2)\). What is \(\overline{A}\cap\overline{B}\)?
Choose one option.
Show answer
- B. \(\{1\}\)
Explanation
\(\overline{A}=[0,1]\) and \(\overline{B}=[1,2]\), whose intersection is \(\{1\}\). Note that \(\overline{A\cap B}=\varnothing\): closure does not commute with intersections.
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-007 Let A be a subset of a topological space X. Which statement is true?
- ZW-CLOSURE-OF-A-SET-009 In a space with the trivial topology \∅,X\, what is the closure of a nonempty subset A?