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Neighbourhood (mathematics)

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.

Definitions

Neighbourhood of a point

Ifis atopological spaceandis a point in, a neighbourhood ofis asubsetofthat includes anopen setcontaining,
This is also equivalent tobeing in theinteriorof.
The neighbourhoodneed not be an open set itself. Ifis open it is called an **open neighbourhood**.[1] Somemathematiciansrequire that neighbourhoods be open, so it is important to note 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 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

Ifis asubsetof topological spacethen a neighbourhood ofis a setthat includes an open setcontaining. It follows that a setis a neighbourhood ofif and only if it is a neighbourhood of all the points in. Furthermore, it follows thatis a neighbourhood ofiffis a subset of theinteriorof. The neighbourhood of a point is just a special case of this definition.

In a metric space

In ametric space, a setis a neighbourhood of a pointif there exists anopen ballwith centreand radius, such that
is contained in.
is called uniform neighbourhood of a setif there exists a positive numbersuch that for all elementsof,
is contained in.
Forthe **-neighbourhood**of a setis the set of all points inthat are at distance less thanfrom(or equivalently,is the union of all the open balls of radiusthat 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

Given the set ofreal numberswith the usualEuclidean metricand a subsetdefined as
thenis a neighbourhood for the setofnatural 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 onis the assignment of afilter(on the set) to eachin, 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 auniform space,is called a uniform neighbourhood ofif there exists anentouragesuch thatcontains all points ofthat 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, theintervalis a neighbourhood ofin thereal line, so the setis 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 thedefinition of the limit of a function.

See also

  • Region (mathematics)

  • Tubular neighbourhood

References

[1]
Citation Linkopenlibrary.orgDixmier, 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 x is nothing more than an open subset of E that contains x.
Sep 29, 2019, 3:52 AM
[2]
Citation Linken.wikipedia.orgThe original version of this page is from Wikipedia, you can edit the page right here on Everipedia.Text is available under the Creative Commons Attribution-ShareAlike License.Additional terms may apply.See everipedia.org/everipedia-termsfor further details.Images/media credited individually (click the icon for details).
Sep 29, 2019, 3:52 AM