Definition:Local Basis

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

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

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 defined as
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 at $x$ is a set $\mathcal B$ of neighborhoods of $x$ such that:
 * $\forall N \subseteq X: \left({\exists U \in \tau: x \in U \subseteq N}\right) \implies \exists H \in \mathcal B: H \subseteq N$

That is, that every neighborhood (open or not, it matters not) of $x$ also contains some set in $\mathcal B$.

Also known as
A local basis is also known as a neighborhood basis.