# Definition:Neighborhood Basis

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## Definition

Let $\struct {S, \tau}$ be a topological space.

Let $x \in S$.

Let $\BB$ be a set of neighborhoods of $x$.

Then $\BB$ is a **neighborhood basis** at $x$ if and only if:

- For each neighborhood $N$ of $x$, there is an $M \in \BB$ 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.

## Also see

- Results about
**neighborhood bases**can be found here.

## Sources

- 1975: Bert Mendelson:
*Introduction to Topology*(3rd ed.) ... (previous) ... (next): Chapter $3$: Topological Spaces: $\S 3$: Neighborhoods and Neighborhood Spaces: Exercise $3$