Exercise · ZW-NEIGHBORHOOD-003 · ★☆☆ Basic
In \(X=\{1,2,3\}\) with \(\mathcal{T}=\{\varnothing,\{1\},\{1,2\},\{1,2,3\}\}\), how many neighborhoods does the point \(3\) have?
Choose one option.
Show answer
- D. 1
Explanation
The only open set containing \(3\) is \(X\), so the only neighborhood is \(X\) itself.
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-001 In X=\1,2,3\ with T=\∅,\1\,\1,2\,\1,2,3\\, which of these sets is a neighborhood of the point 2?
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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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