Exercise · ZW-INTERIOR-OF-A-SET-010 · ★★★ Advanced
Which of these equalities is false in general?
Choose one option.
Show answer
- B. \(\operatorname{int}(A\cup B)=\operatorname{int}A\cup\operatorname{int}B\)
Explanation
Only «\(\supseteq\)» holds for unions. With \(X=\{1,2,3\}\), \(\mathcal{T}=\{\varnothing,\{1\},\{1,2\},X\}\), \(A=\{2\}\) and \(B=\{1,3\}\): \(\operatorname{int}(A\cup B)=X\), but \(\operatorname{int}A\cup\operatorname{int}B=\{1\}\).
Review the concept: Interior of a set
The interior \(\operatorname{int}A\) of a subset \(A\) of a topological space \((X,\mathcal{T})\) is the largest open set \(U\in\mathcal{T}\) contained in \(A\).
\[\operatorname{int}A=\bigcup\{U\in\mathcal{T}:\ U\subseteq A\}\]
More practice problems: interior of a set
- ZW-INTERIOR-OF-A-SET-001 In X=\1,2,3\ with T=\∅,\1\,\1,2\,\1,2,3\\, what is the interior of A=\1,3\?
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