Exercise · ZW-ACCUMULATION-POINT-008 · ★★☆ Intermediate
In usual \(\mathbb{R}\), what is the derived set \(\mathbb{Q}'\) of the rationals?
Choose one option.
Show answer
- D. \(\mathbb{R}\)
Explanation
Every open interval around any real number contains infinitely many rationals different from that number (the rationals are dense). So every real is an accumulation point of \(\mathbb{Q}\) and \(\mathbb{Q}'=\mathbb{R}\).
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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