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 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 $$\mathcal L$$.