Exercise · ZW-ACCUMULATION-POINT-009 · ★★★ Advanced
Let \(X\) have the discrete topology and \(A\subseteq X\) arbitrary. What is \(A'\)?
Choose one option.
Show answer
- B. \(\varnothing\)
Explanation
In the discrete topology \(\{x\}\) is open, it is a neighborhood of \(x\), and it contains no points of \(A\) other than \(x\). So no point is an accumulation point and \(A'=\varnothing\).
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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