Neighbourhood basis

Neighbourhood basis

A neighbourhood basis or local basis (or neighbourhood base or local base) for a point ${\displaystyle x}$ is a filter base of the neighbourhood filter; this means that it is a subset $${\displaystyle {\mathcal {B}}\subseteq {\mathcal {N}}(x)}$$such that for all ${\displaystyle V\in {\mathcal {N}}(x),}$ there exists some ${\displaystyle B\in {\mathcal {B}}}$ such that ${\displaystyle B\subseteq V.}$ Here, ${\displaystyle {\mathcal {N}}(x)}$ denotes the set of all neighbourhoods of ${\displaystyle x}$. That is, for any neighbourhood ${\displaystyle V}$ we can find a neighbourhood ${\displaystyle B}$ in the neighbourhood basis that is contained in ${\displaystyle V.}$