Neighbourhood (mathematics)

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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.

A set in the plane is a neighbourhood of a point if a small disc around is contained in

DefinitionsEdit

Neighbourhood of a pointEdit

If   is a topological space and   is a point in   then 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 does not have a neighbourhood on any of its corners or its boundary since there is no open set containing any corner.

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.

Neighbourhood of a setEdit

If   is a subset of a topological space  , then a neighbourhood of   is a set   that includes an open set   containing  ,

 
It follows that a set   is a neighbourhood of   if and only if it is a neighbourhood of all the points in   Furthermore,   is a neighbourhood of   if and only if   is a subset of the interior of   A neighbourhood of   that is also an open subset of   is called an open neighbourhood of   The neighbourhood of a point is just a special case of this definition.

In a metric spaceEdit

 
A set   in the plane and a uniform neighbourhood   of  
 
The epsilon neighbourhood of a number   on the real number line.

In a metric space   a set   is a neighbourhood of a point   if there exists an open ball with center   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 centered 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  

ExamplesEdit

 
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 neighbourhoodsEdit

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, 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.

Uniform neighbourhoodsEdit

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 neighbourhoodEdit

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).[2]

See alsoEdit

ReferencesEdit

  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  
  2. ^ Peters, Charles (2022). "Professor Charles Peters" (PDF). University of Houston Math. Retrieved 3 April 2022.{{cite web}}: CS1 maint: url-status (link)