Exercise · ZW-ACCUMULATION-POINT-004 · ★☆☆ Basic
In \(X=\{a,b,c\}\) with \(\mathcal{T}=\{\varnothing,\{a,b,c\}\}\), what is the derived set of \(A=\{a\}\)?
Choose one option.
Show answer
- C. \(\{b,c\}\)
Explanation
In a finite space it suffices to check the smallest open set containing each point. \(a\): the smallest open set is \(\{a,b,c\}\), which contains no points of \(A\) other than \(a\) (not in \(A'\)); \(b\): the smallest open set is \(\{a,b,c\}\), which contains points of \(A\) other than \(b\) (it is in \(A'\)); \(c\): the smallest open set is \(\{a,b,c\}\), which contains points of \(A\) other than \(c\) (it is in \(A'\)). Hence \(A'=\{b,c\}\).
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
- ZW-ACCUMULATION-POINT-001 In X=\1,2,3\ with T=\∅,\1\,\1,2\,\1,2,3\\, what is the derived set of A=\1,3\?
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