In \(X=\{1,2,3,4\}\) with \(\mathcal{T}=\{\varnothing,\{1\},\{3\},\{1,3\},\{1,2,3\},\{1,3,4\},\{1,2,3,4\}\}\), which is the smallest neighborhood of the point \(2\)?
The open sets containing \(2\) are \(\{1,2,3\}\) and \(X\); the smaller one is \(\{1,2,3\}\). Neither \(\{2\}\) nor \(\{1,2\}\) contains an open set with \(2\).
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.
\[N\ \text{neighborhood of }x\iff \exists\,U\in\mathcal{T}:\ x\in U\subseteq N\]