Definition:Local Basis

Definition
Let $T = \left({S, \vartheta}\right)$ be a topological space.

Let $x \in S$ be a point in $S$.

A local basis (or neighborhood basis) at $x$ is a set $\mathcal B$ of open neighborhoods of $x$ such that:
 * $\forall U \in \vartheta: x \in U \implies \exists H \in \mathcal B: H \subseteq U$

That is, that every open set of $S$ containing $x$ also contains at least one of the sets of $\mathcal B$.