Exercise · ZW-NEIGHBORHOOD-008 · ★★☆ Intermediate
Which statement about the neighborhoods of a point \(x\) is true?
Choose one option.
Show answer
- D. If \(N\) is a neighborhood of \(x\) and \(N\subseteq M\), then \(M\) is one too.
Explanation
If \(U\) is open with \(x\in U\subseteq N\), then also \(U\subseteq M\). Neighborhoods need not be open, and there are usually many of them.
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-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-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…