In topology and related areas of mathematics, the neighbourhood system or neighbourhood filter 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 we can find a neighbourhood in the neighbourhood basis which is contained in .
Conversely, as with any filter base, the local basis allows to get back the corresponding neighbourhood filter as .
Read more about Neighbourhood System: Examples, Properties
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