Definition:Basis for Neighborhood System

Definition
Let $M = \left({A, d}\right)$ be a metric space.

Let $a \in A$ be a point in $M$.

Let $\mathcal B_a$ be a set of neighborhoods of $a$ in $M$.

Then $\mathcal B_a$ is a basis for the neighborhood system at $a$ iff:
 * $\forall N_a \subseteq M: \exists B \in \mathcal B_a: B \subseteq N_a$

where $N_a$ denotes a neighborhood of $a$ in $M$.

That is, $\mathcal B_a$ is a basis for the neighborhood system at $a$ iff every neighborhood of $a$ contains an element of $\mathcal B_a$ as a subset.