Exercise · ZW-ACCUMULATION-POINT-005 · ★☆☆ Basic
In \(X=\{1,2,3\}\) with \(\mathcal{T}=\{\varnothing,\{2\},\{1,2\},\{2,3\},\{1,2,3\}\}\), what is the derived set of \(A=\{2,3\}\)?
Choose one option.
Show answer
- B. \(\{1,3\}\)
Explanation
In a finite space it suffices to check the smallest open set containing each point. \(1\): the smallest open set is \(\{1,2\}\), which contains points of \(A\) other than \(1\) (it is in \(A'\)); \(2\): the smallest open set is \(\{2\}\), which contains no points of \(A\) other than \(2\) (not in \(A'\)); \(3\): the smallest open set is \(\{2,3\}\), which contains points of \(A\) other than \(3\) (it is in \(A'\)). Hence \(A'=\{1,3\}\).
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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