Neighbourhood (mathematics)

In topology and related areas of mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a topological space.

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

Intuitively speaking, a neighbourhood of a point is a set of points containing that point where one can move some amount in any direction away from that point without leaving the set.

that includes an open set

belonging to the topological interior of

it is called an open neighbourhood[2] (resp.

Some authors[3] require neighbourhoods to be open, so it is important to note their conventions.

A set that is a neighbourhood of each of its points is open since it can be expressed as the union of open sets containing each of its points.

A closed rectangle, as illustrated in the figure, is not a neighbourhood of all its points; points on the edges or corners of the rectangle are not contained in any open set that is contained within the rectangle.

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

is a subset of a topological space

that includes an open set

is called an open neighbourhood of

The neighbourhood of a point is just a special case of this definition.

if there exists an open ball with center

is called a uniform neighbourhood of a set

if there exists a positive number

is the union of all the open balls of radius

-neighbourhood is a uniform neighbourhood, and that a set is a uniform neighbourhood if and only if it contains an

Given the set of real numbers

with the usual Euclidean metric and a subset

of natural numbers, but is not a uniform neighbourhood of this set.

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.

such that One can show that both definitions are compatible, that is, 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.

is called a uniform neighbourhood of

A deleted neighbourhood of a point

in the real line, so the set

A deleted neighbourhood of a given point is not in fact a neighbourhood of the point.

The concept of deleted neighbourhood occurs in the definition of the limit of a function and in the definition of limit points (among other things).