Exercise · ZW-NEIGHBORHOOD-006 · ★☆☆ Basic
In \(X=\{a,b,c\}\) with \(\mathcal{T}=\{\varnothing,\{a,b,c\}\}\), how many neighborhoods does the point \(a\) have?
Choose one option.
Show answer
- B. 1
Explanation
In the trivial topology the only nonempty open set is \(X\), so the only neighborhood of any point is \(X\).
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?
- 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…
- ZW-NEIGHBORHOOD-005 In X=\1,2,3,4\ with T=\∅,\1\,\3\,\1,3\,\1,2,3\,\1,3,4\,\1,2,3,4\\, which is the smallest neighborho…
- 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?
- ZW-NEIGHBORHOOD-009 In X=\1,2,3,4\ with T=\∅,\1\,\3\,\1,3\,\1,2,3\,\1,3,4\,\1,2,3,4\\, of which points of X is \1,2\ a…