Definition:Local Basis

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

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

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

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