Exercise · ZW-CLOSURE-OF-A-SET-001 · ★☆☆ Basic
In \(X=\{1,2,3\}\) with \(\mathcal{T}=\{\varnothing,\{1\},\{1,2\},\{1,2,3\}\}\), what is the closure of \(\{3\}\)?
Choose one option.
Show answer
- C. \(\{3\}\)
Explanation
The closed sets containing \(\{3\}\) are \(\{3\}\), \(\{2,3\}\), \(\{1,2,3\}\); their intersection is \(\{3\}\).
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-002 In X=\1,2,3\ with T=\∅,\1\,\1,2\,\1,2,3\\, what is the closure of \2\?
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