\begin{definition}[Definition:Local Basis/Local Basis for Open Sets]
Let $T = \struct {S, \tau}$ be a topological space.
Let $x$ be an element of $S$.
A '''local basis''' at $x$ is a set $\BB$ of open neighborhoods of $x$ such that:
:$\forall U \in \tau: x \in U \implies \exists H \in \BB: H \subseteq U$
That is, such that every open neighborhood of $x$ also contains some set in $\BB$.
\end{definition}