Neighbourhood system


In topology and related areas of mathematics, the neighbourhood system, complete system of neighbourhoods, or neighbourhood filter for a point x is the collection of all neighbourhoods of the point x.

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 we can find a neighbourhood in the neighbourhood basis that is contained in.
Conversely, as with any filter base, the local basis allows the corresponding neighbourhood filter to be recovered as.
A neighbourhood subbasis at x is a collection ? of subsets of X, each of which contains x, such that the collection of all possible finite intersections of elements of ? forms a neighborhood basis at x.

Examples

In a seminormed space, that is a vector space with the topology induced by a seminorm, all neighbourhood systems can be constructed by translation of the neighbourhood system for the point 0,
This is because, by assumption, vector addition is separately 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.