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 of these sets is a neighborhood of at least one point?
Only \(\{1,2\}\) contains a nonempty open set (\(\{1\}\)). The sets \(\{2\}\), \(\{4\}\) and \(\{2,4\}\) contain no nonempty open set, so they are neighborhoods of no 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.
\[N\ \text{neighborhood of }x\iff \exists\,U\in\mathcal{T}:\ x\in U\subseteq N\]