Exercise · ZW-ACCUMULATION-POINT-001 · ★☆☆ Basic

In \(X=\{1,2,3\}\) with \(\mathcal{T}=\{\varnothing,\{1\},\{1,2\},\{1,2,3\}\}\), what is the derived set of \(A=\{1,3\}\)?

Choose one option.

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  • B. \(\{2,3\}\)

Explanation

In a finite space it suffices to check the smallest open set containing each point. \(1\): the smallest open set is \(\{1\}\), which contains no points of \(A\) other than \(1\) (not in \(A'\)); \(2\): the smallest open set is \(\{1,2\}\), which contains points of \(A\) other than \(2\) (it is in \(A'\)); \(3\): the smallest open set is \(\{1,2,3\}\), which contains points of \(A\) other than \(3\) (it is in \(A'\)). Hence \(A'=\{2,3\}\).

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Review the concept: Accumulation point

A point \(x\) of a topological space \(X\) is an accumulation point of a subset \(A\) if every neighborhood \(N\) of \(x\) contains a point of \(A\) other than \(x\); these points form the derived set \(A'\).

\[x\in A'\iff \forall N\text{ neighborhood of }x:\ (N\setminus\{x\})\cap A\neq\varnothing\]

More practice problems: accumulation point

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