Exercise · ZW-ACCUMULATION-POINT-002 · ★☆☆ Basic

In \(X=\{a,b,c,d\}\) with \(\mathcal{T}=\{\varnothing,\{a\},\{b\},\{a,b\},\{a,b,c\},\{a,b,c,d\}\}\), what is the derived set of \(A=\{a,c\}\)?

Choose one option.

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  • D. \(\{c,d\}\)

Explanation

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

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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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