Exercise · ZW-INTERIOR-OF-A-SET-009 · ★★★ Advanced

In \(X=\{1,2,3,4\}\) with \(\mathcal{T}=\{\varnothing,\{1\},\{3\},\{1,3\},\{1,2,3\},\{1,3,4\},\{1,2,3,4\}\}\), how many subsets \(A\subseteq X\) have empty interior?

Choose one option.

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  • C. 4

Explanation

The interior is empty iff \(A\) contains no nonempty open set, that is, iff it contains neither \(1\) nor \(3\) (both are open singletons). These are the subsets of \(\{2,4\}\): 4.

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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

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