Exercise · ZW-ACCUMULATION-POINT-007 · ★★☆ Intermediate
In usual \(\mathbb{R}\), let \(A=\{1/n:\ n\in\mathbb{N},\ n\ge 1\}\). What is \(A'\)?
Choose one option.
Show answer
- A. \(\{0\}\)
Explanation
The points \(1/n\) are isolated: around \(1/n\) there is an interval containing no other \(1/m\). By contrast, every interval around \(0\) contains infinitely many \(1/n\), so \(0\) is the only accumulation point. Note that \(0\notin A\).
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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