Definition:Local Basis/Local Basis for Open Sets

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

Let $x$ be an element of $S$.

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

That is, such that every open neighborhood of $x$ also contains some set in $\mathcal B$.

Also see

 * Equivalence of Definitions of Local Basis