Neighbourhood (mathematics)

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A set
V
{\displaystyle V}
in the plane is a neighbourhood of a point
p
{\displaystyle p}
if a small disc around
p
{\displaystyle p}
is contained in
V
{\displaystyle V}
. Neighborhood illust1.svg
A set in the plane is a neighbourhood of a point if a small disc around is contained in .

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.

Contents

Definitions

Neighbourhood of a point

If is a topological space and is a point in , a neighbourhood of is a subset of that includes an open set containing

This is also equivalent to the point belonging to the topological interior of in

The neighbourhood need not be an open subset but when is open in then it is called an open neighbourhood. [1] Some authors have been known to require neighbourhoods to be open, so it is important to note conventions.

A closed rectangle is not a neighbourhood of any of its corners (or points on the boundary). Neighborhood illust2.svg
A closed rectangle is not a neighbourhood of any of its corners (or points on the boundary).

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 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.

Neighbourhood of a set

If is a subset of topological space then a neighbourhood of is a set that includes an open set containing . 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. Furthermore, V is a neighbourhood of S if and only if S is a subset of the interior of V. A neighbourhood of S that is also an open set is called an open neighbourhood of S. The neighbourhood of a point is just a special case of this definition.

In a metric space

A set
S
{\displaystyle S}
in the plane and a uniform neighbourhood
V
{\displaystyle V}
of
S
{\displaystyle S}
. Neighborhood illust3.svg
A set in the plane and a uniform neighbourhood of .
The epsilon neighbourhood of a number a on the real number line. Epsilon Umgebung.svg
The epsilon neighbourhood of a number a on the real number line.

In a metric space , a set is a neighbourhood of a point if there exists an open ball with centre and radius , such that

is contained in .

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

is contained in .

For the -neighbourhood of a set is the set of all points in that are at distance less than from (or equivalently, is the union of all the open balls of radius that are centred at a point in ):

It directly follows that an -neighbourhood is a uniform neighbourhood, and that a set is a uniform neighbourhood if and only if it contains an -neighbourhood for some value of .

Examples

The set M is a neighbourhood of the number a, because there is an e-neighbourhood of a which is a subset of M. Set of real numbers with epsilon-neighbourhood.svg
The set M is a neighbourhood of the number a, because there is an ε-neighbourhood of a which is a subset of M.

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

then is a neighbourhood for the set 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 neighbourhood system on is the assignment of a filter of subsets of to each in , such that

  1. the point is an element of each in
  2. each in contains some in such that for each in , is in .

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 , is called a uniform neighbourhood of if there exists an entourage such that contains all points of that are -close to some point of ; that is, for all .

Deleted neighbourhood

A deleted neighbourhood of a point (sometimes called a punctured neighbourhood) is a neighbourhood of , without . For instance, the interval is a neighbourhood of in the real line, so the set is a deleted neighbourhood of . 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).

See also

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References

  1. Dixmier, Jacques (1984). General Topology . Undergraduate Texts in Mathematics. Translated by Sterling K. Berberian. Springer. p.  6. ISBN   0-387-90972-9. According to this definition, an open neighborhood of is nothing more than an open subset of that contains