\begin{definition}[Definition:Neighborhood System]
Let $T = \struct {S, \tau}$ be a topological space.
A '''neighborhood system''' is a family $\family {\NN_x}_{x \mathop \in S}$ indexed by points of $S$, such that $\NN_x$ is a local basis at $x$ for $x \in S$.
\end{definition}