Exercise · ZW-ACCUMULATION-POINT-010 · ★★★ Advanced
In usual \(\mathbb{R}\), let \(A=(0,1]\cup\{2\}\). What is \(A'\)?
Choose one option.
Show answer
- C. \([0,1]\)
Explanation
The points of \([0,1]\) are accumulation points (including \(0\), which is not in \(A\), and \(1\)). The point \(2\) belongs to \(A\) but is isolated: \((1.5,2.5)\) contains no other points of \(A\). So \(A'=[0,1]\), whereas \(\overline{A}=A\cup A'\) does include \(2\).
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\?
- ZW-ACCUMULATION-POINT-002 In X=\a,b,c,d\ with T=\∅,\a\,\b\,\a,b\,\a,b,c\,\a,b,c,d\\, what is the derived set of A=\a,c\?
- ZW-ACCUMULATION-POINT-003 In X=\1,2,3,4\ with T=\∅,\1\,\3\,\1,3\,\1,2,3\,\1,3,4\,\1,2,3,4\\, what is the derived set of A=\1,…
- ZW-ACCUMULATION-POINT-004 In X=\a,b,c\ with T=\∅,\a,b,c\\, what is the derived set of A=\a\?
- ZW-ACCUMULATION-POINT-005 In X=\1,2,3\ with T=\∅,\2\,\1,2\,\2,3\,\1,2,3\\, what is the derived set of A=\2,3\?
- ZW-ACCUMULATION-POINT-006 In ℝ with the usual topology, what is the derived set of A=(0,1)?
- ZW-ACCUMULATION-POINT-007 In usual ℝ, let A=\1/n:\ n∈ℕ,\ n≥ 1\. What is A'?
- ZW-ACCUMULATION-POINT-008 In usual ℝ, what is the derived set ℚ' of the rationals?