# Definition:Neighborhood Basis

## Definition

Let $\struct{X, \tau}$ 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 $x$ if and only if:

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 $\mathsf{Pr} \infty \mathsf{fWiki}$, such a structure is referred to as a local basis.

## Also known as

Some sources call this a local basis, but on $\mathsf{Pr} \infty \mathsf{fWiki}$, that term is reserved for a stronger notion.