In topology and related areas of mathematics, the neighbourhood system, complete system of neighbourhoods, or neighbourhood filter
Contents
Basis
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
Conversely, as with any filter base, the local basis allows to get back the corresponding neighbourhood filter as
Examples
Properties
In a semi normed space, that is a vector space with the topology induced by a semi norm, all neighbourhood systems can be constructed by translation of the neighbourhood system for the point 0,
This is because, by assumption, vector addition is separate continuous in the induced topology. Therefore, the topology is determined by its neighbourhood system at the origin. More generally, this remains true whenever the space is a topological group or the topology is defined by a pseudometric.
Every neighbourhood system for a non empty set A is a filter called the neighbourhood filter for A.