Exercise · ZW-NEIGHBORHOOD-001 · ★☆☆ Basic
In \(X=\{1,2,3\}\) with \(\mathcal{T}=\{\varnothing,\{1\},\{1,2\},\{1,2,3\}\}\), which of these sets is a neighborhood of the point \(2\)?
Choose one option.
Show answer
- C. \(\{1,2\}\)
Explanation
It contains the open set \(\{1,2\}\), which in turn contains \(2\). The others contain no open set having \(2\) as a member.
Review the concept: Neighborhood of a point
A neighborhood of a point \(x\) in a topological space \((X,\mathcal{T})\) is a set \(N\subseteq X\) that contains some open set \(U\in\mathcal{T}\) to which \(x\) belongs.
\[N\ \text{neighborhood of }x\iff \exists\,U\in\mathcal{T}:\ x\in U\subseteq N\]
More practice problems: neighborhood of a point
- ZW-NEIGHBORHOOD-002 In X=\1,2,3\ with T=\∅,\1\,\1,2\,\1,2,3\\, which is the smallest neighborhood of the point 1?
- ZW-NEIGHBORHOOD-003 In X=\1,2,3\ with T=\∅,\1\,\1,2\,\1,2,3\\, how many neighborhoods does the point 3 have?
- ZW-NEIGHBORHOOD-004 In X=\a,b,c,d\ with T=\∅,\a\,\b\,\a,b\,\a,b,c\,\a,b,c,d\\, how many subsets of X are neighborhoods…
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- ZW-NEIGHBORHOOD-006 In X=\a,b,c\ with T=\∅,\a,b,c\\, how many neighborhoods does the point a have?
- ZW-NEIGHBORHOOD-007 Let U be an open set of X and x∈ U. Which statement is correct?
- ZW-NEIGHBORHOOD-008 Which statement about the neighborhoods of a point x is true?
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