Definition:Neighborhood Basis

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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.