Neighbourhood (topology): Difference between revisions

From Citizendium
Jump to navigation Jump to search
imported>Peter Schmitt
(transient)
imported>Milton Beychok
m (Deleted "See Also" section which is a WP-ism not used in CZ. Created Metadata template. Reformatted "External Links" subpage.)
Line 43: Line 43:
: ''(Definition)'' <br> In a neighbourhood space, a set is '''open''' if it is a neighbourhood of all its points.
: ''(Definition)'' <br> In a neighbourhood space, a set is '''open''' if it is a neighbourhood of all its points.
: ''(Definition)'' <br> In a topological space, a set is a '''neighbourhood''' of a point if it contains an open set that contains the point. <br> (In other words, the open sets containing a point form a '''base''' for the neighbourhood system of this point.)
: ''(Definition)'' <br> In a topological space, a set is a '''neighbourhood''' of a point if it contains an open set that contains the point. <br> (In other words, the open sets containing a point form a '''base''' for the neighbourhood system of this point.)
==See also==
* [[Topological space#Some topological notions]]

Revision as of 18:11, 27 May 2009

This article has a Citable Version.
Main Article
Discussion
Related Articles  [?]
Bibliography  [?]
External Links  [?]
Citable Version  [?]
 
This editable Main Article has an approved citable version (see its Citable Version subpage). While we have done conscientious work, we cannot guarantee that this Main Article, or its citable version, is wholly free of mistakes. By helping to improve this editable Main Article, you will help the process of generating a new, improved citable version.

In topology, the notion of a neighbourhood is used to describe, in an abstract setting, the concept of points near a given point. It is modelled after the situation in real analysis where the points in small balls are considered as near to the centre of the ball.

Neighbourhoods are used to define convergence and continuous functions:

Convergence (Definition)
A sequence converges to a point if and only if every neighbourhood of that point contains almost all (i.e., all but finitely many) elements of the sequence.
Continuity (Definition)
A function f is continuous at a point x if and only if for every neighbourhood U of f(x) there is a neighbourhood V of x for which the image f(V) under f is a subset of U.

Neighbourhood spaces

A set X is called a neighbourhood space if for every x in X there is a nonempty family N(x) of sets, called neighbourhoods of x, which satisfies the following axioms:

  1. x is an element of every neighborhood of x.
  2. Any set that contains a neighbourhood of x is a neighbourhood of x.
  3. The intersection of any two (and therefore of any finite collection of) neighbourhoods of x is a neighbourhood of x.
  4. Any neighbourhood of x contains an open neighbourhood of x,
    i.e., a neighbourhood of x that belongs to N(y) for all of its elements y.

Axioms (1-3) imply that N(x) is a filter. Accordingly, the system of neighbourhoods of a point is also called the neighbourhood filter of the point.
Axiom (4) defines how neighbourhood systems of distinct points interact.

Neighbourhood base

To define a neighbourhood space it is often more convenient to describe only a base for the neighbourhood system. A nonempty family B(x) of sets is a neighbourhood base if it satisfies the following axioms:

  1. x is an element of every set in B(x).
  2. The intersection of any two sets of B(x) contains a set of B(x).
  3. Any neighbourhood of x contains an open neighbourhood of x,
    i.e., a neighbourhood of x that belongs to N(y) for all of its elements y.

Relation to topological spaces

Neighbourhood spaces are one of several equivalent means to define a topological space. The equivalence is obtained by the following definitions:

(Definition)
In a neighbourhood space, a set is open if it is a neighbourhood of all its points.
(Definition)
In a topological space, a set is a neighbourhood of a point if it contains an open set that contains the point.
(In other words, the open sets containing a point form a base for the neighbourhood system of this point.)