Local base

In topology and related areas of mathematics, the neighbourhood system, complete system of neighbourhoods, or neighbourhood filter ${\displaystyle {\mathcal {V}}(x)}$ for a point x is the collection of all neighbourhoods for the point x.

A neighbourhood basis or local basis for a point x is a filter base of the neighbourhood filter, i.e. a subset

such that

That is, for any neighbourhood ${\displaystyle V}$ we can find a neighbourhood ${\displaystyle B}$ in the neighbourhood basis that is contained in ${\displaystyle V}$.

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