Neighbourhood (mathematics)

In topology and related areas of mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a topological space. Intuitively speaking, a neighbourhood of a point is a set containing the point where you can "wiggle" or "move" the point a bit without leaving the set. In other words, a neighbourhood contains an open set which contains the point. The collection of all neighbourhoods for a point is called the neighbourhood system for the point.

In a metric space where we have a notion of distance this concept is straightforward to define, but it can be generalized to more abstract topological spaces with no concrete way to measure distances.

This concept is closely related to the concepts of open set and interior.

Definition

In a metric space M:=(X,d), a set V is called a neighbourhood for a point p if there exists an open ball with radius r centred at p

which is contained in V.

A set V is called a neighbourhood for a set P if V is a neighbourhood for all elements of P.

V is called uniform neighbourhood for a set P if there exists r a positive number such that for all elements of p of P

is contained in V.

Examples

Given the set of real numbers R with the usual Euclidean metric and a subset V defined as

then V is a neighbourhood for the set N of natural numbers, but is not a uniform neighbourhood of this set.

Generalized definition

For an arbitrary topological space X, V is called neighbourhood for p if p is an element of the interior of V.

In a uniform space S:=(X, δ) V is called a uniform neighbourhood of P if P is not close to X \ V, that is there exists no entourage containing P and X \ V.