Definition:Basis for Neighborhood System

Definition
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 $a$ :
 * $\forall N_a \subseteq M: \exists B \in \BB_a: B \subseteq N_a$

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

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