Exercise · ZW-NEIGHBORHOOD-002 · ★☆☆ Basic
In \(X=\{1,2,3\}\) with \(\mathcal{T}=\{\varnothing,\{1\},\{1,2\},\{1,2,3\}\}\), which is the smallest neighborhood of the point \(1\)?
Choose one option.
Show answer
- A. \(\{1\}\)
Explanation
The set \(\{1\}\) is open, contains \(1\), and is contained in every other neighborhood.
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
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