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 $\mathcal B$.

Alternative Definition
Some more modern sources suggest that in order to be a local basis, the neighborhoods of which the set $\mathcal B$ consists of do not need to be open.

With this condition, the definition goes:

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

That is, that every neighborhood (closed or open, it matters not) of $x$ also contains at least one of the sets of $\mathcal B$.