neighbourhood system

A neighbourhood system is another name for a neighbourhood filter.

Therefore the topology is determined by its neighbourhood system at the origin.

The collection of all neighbourhoods of a point is called the neighbourhood system at the point.

Trivially the neighbourhood system for a point is also a neighbourhood basis for the point.

In topology and related areas of mathematics, a neighbourhood space is a set X such that for each there is an associated neighbourhood system .

Every neighbourhood system for a non empty set A is a filter called the neighbourhood filter for A.

In a metric space, it is equivalent to consider the neighbourhood system of open balls centered at 'x' and 'f'('x') instead of all neighborhoods.

So a small neighbourhood system often works best if service users know that they can, if they wish, choose to be treated in a neighbouring locality by another consultant.

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.

There is an alternative way to define a topology, by first defining the neighbourhood system, and then open sets as those sets containing a neighbourhood of each of their points.

For example, much publicity has been given to the patch or neighbourhood system of social work which entails a small integrated team of social workers dealing ideally with small communities of between 5,000 and 10,000 people.

One can show that both definitions are compatible, i.e. the topology obtained from the neighbourhood system defined using open sets is the original one, and vice versa when starting out from a neighbourhood system.