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.

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

Definition

For an arbitrary topological space X, V is called neighbourhood of a point p (or a subset S), if p is an element of (or S is contained in) some open set contained in V.

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

The case of a metric space

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

<math>B_r(p) = B(p;r) = \{ x \in X \mid d(x,p) < r \}<math>

which is contained in V.

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

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

<math>B_r(p) = \{ x \in X \mid d(x,p) < r \}<math>

is contained in V.

Examples

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

<math>V:=\bigcup_{n \in \mathbb{N}} B\big(n\,;\,\frac{1}{n}\big),<math>

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

Topology from neigborhoods

The above definition is useful if the notion of open set is already defined. 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.

A neighborhood system on X is the assignment of a filter N(x) (on the set X) to each x in X, such that

1. the point x is an element of each U in N(x)
2. each U in N(x) contains some V in N(x) such that for each y in V, U is in N(y).

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.

Generalized definition

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.