Neighbourhood system
From Exampleproblems
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 V we can find a neighbourhood B in the neighbourhood basis which is contained in V.
Conversely, as with any filter base, the local basis allows to get back the corresponding neighbourhood filter as 
.
Examples
- Trivially the neighbourhood system for a point is also a neighbourhood basis for the point.
- Given a space X with the indiscrete topology the neighbourhood system for any point x is the whole space,
- In a metric space, for any point x, the sequence of open balls around x with radius 1/n form a countable neighbourhood basis
. This means every metric space is first-countable.
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,
More generally, this remains true whenever the topology is defined by a translation invariant metric or pseudometric.
Every neighbourhood system for a non empty set A is a filter called the neighbourhood filter for A.
The union of local bases for all points x are a base for the topology.
