Definition:Neighborhood Basis

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

Let $x \in X$.

Let $\mathcal B$ be a set of neighborhoods of $x$.

Then $\mathcal B$ is a neighborhood basis at $x$ iff:
 * For each neighborhood $N$ of $x$, there is an $M \in \mathcal B$ such that $M \subseteq N$.

Also defined as
Some sources require the elements of a neighborhood basis to be open.

On, such a structure is referred to as a local basis.

Also known as
Some sources call this a local basis, but on, that term is reserved for a stronger notion.

Also see

 * Local Basis is Neighborhood Basis
 * Definition:Neighborhood Sub-Basis