In \(X=\{a,b,c,d\}\) with \(\mathcal{T}=\{\varnothing,\{a\},\{b\},\{a,b\},\{a,b,c\},\{a,b,c,d\}\}\), how many subsets of \(X\) are neighborhoods of the point \(c\)?
The open sets containing \(c\) are \(\{a,b,c\}\) and \(X\); the neighborhoods are the sets containing one of them: \(\{a,b,c\}\) and \(X\), that is, 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.
\[N\ \text{neighborhood of }x\iff \exists\,U\in\mathcal{T}:\ x\in U\subseteq N\]