Exercise · ZW-ACCUMULATION-POINT-006 · ★★☆ Intermediate
In \(\mathbb{R}\) with the usual topology, what is the derived set of \(A=(0,1)\)?
Choose one option.
Show answer
- C. \([0,1]\)
Explanation
Every point of \((0,1)\) is an accumulation point because every interval around it contains other points of \((0,1)\); also \(0\) and \(1\) are, since every interval around them meets \((0,1)\) in points other than themselves. So \(A'=[0,1]\).
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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