Exercise · ZW-ACCUMULATION-POINT-003 · ★☆☆ Basic
In \(X=\{1,2,3,4\}\) with \(\mathcal{T}=\{\varnothing,\{1\},\{3\},\{1,3\},\{1,2,3\},\{1,3,4\},\{1,2,3,4\}\}\), what is the derived set of \(A=\{1,2\}\)?
Choose one option.
Show answer
- A. \(\{2,4\}\)
Explanation
In a finite space it suffices to check the smallest open set containing each point. \(1\): the smallest open set is \(\{1\}\), which contains no points of \(A\) other than \(1\) (not in \(A'\)); \(2\): the smallest open set is \(\{1,2,3\}\), which contains points of \(A\) other than \(2\) (it is in \(A'\)); \(3\): the smallest open set is \(\{3\}\), which contains no points of \(A\) other than \(3\) (not in \(A'\)); \(4\): the smallest open set is \(\{1,3,4\}\), which contains points of \(A\) other than \(4\) (it is in \(A'\)). Hence \(A'=\{2,4\}\).
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-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?
- ZW-ACCUMULATION-POINT-009 Let X have the discrete topology and A⊆ X arbitrary. What is A'?