Neighbourhood (mathematics)

See also the concepts of neighbour and neighbourhood in graph theory.

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

If X is a topological space and p is a point in X, a neighbourhood of p is a set V, which contains an open set U containing p.

p \in U \subseteq V

Note that the neighbourhood V need not be an open set itself. If V is open it is called an open neighbourhood. Some authors require that neighbourhoods be open; be careful to note conventions.

If S is a subset of X, a neighbourhood of S is a set V, which contains an open set U containing S. It follows that a set V is a neighbourhood of S, if and only if, it is a neighbourhood of all the points in S.

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

In a metric space

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

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

which is contained in V.

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

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

is contained in V.

Examples

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

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

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

Topology from neighbourhoods

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

the point x is an element of each U in N(x)
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.

Uniform neighbourhoods

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.

Significance of neighbourhoods in analysis of real functions

Neighbourhoods in more than one dimension are generally chosen equivalent to Euclidean metrics for their symmetry and readability. But for the analysis of functions, this choice is wholly arbitrary. Other notions of distance will (as they ought to) lead to the same results in analysis, if they are properly formulated. Consider, for instance, the following notion of distance in two dimensions: let

(x,y)

be in the

\delta

-neighbourhood of

(x_0,y_0)

if and only if

|x - x_0| + |y - y_0| < \delta,\,

This figure is a square centered on $(x_0,y_0)$ and inscribed in the circle of radius $\delta$ centered there such that its vertices lie exactly on the east, north, west, and south of the circle. Obviously, then, this neighbourhood is a subset of a standard neighbourhood, and all the results in analysis that hold for standard neighbourhoods also hold for these squares (since analysis is concerned with selecting arbitrarily small $\delta$ s).

The conclusion is that there is nothing special about the shape we assign to a neighbourhood in multiple dimensions. The important (distinguishing) requirement for neighbourhoods in analysis is much looser. See metric (mathematics) for a more thorough discussion.