\begin{definition}[Definition:Basis for Neighborhood System]
Let $M = \struct {A, d}$ be a metric space.
Let $a \in A$ be a point in $M$.
Let $\BB_a$ be a set of neighborhoods of $a$ in $M$.
Then $\BB_a$ is a '''basis for the neighborhood system at $a$''' {{iff}}:
:$\forall N_a \subseteq M: \exists B \in \BB_a: B \subseteq N_a$
where $N_a$ denotes a neighborhood of $a$ in $M$.
That is, $\BB_a$ is a '''basis for the neighborhood system at $a$''' {{iff}} every neighborhood of $a$ contains an element of $\BB_a$ as a subset.
\end{definition}