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{{for|the concept in graph theory|Neighbourhood (graph theory)}} | |||
[[File:Neighborhood illust1.png|right|thumb|A set <math>V</math> in the [[plane (geometry)|plane]] is a neighbourhood of a point <math>p</math> if a small disk around <math>p</math> is contained in <math>V</math>.]] | |||
[[File:Neighborhood illust2.svg|right|thumb|A rectangle is not a neighbourhood of any of its corners.]] | |||
In [[topology]] and related areas of [[mathematics]], a '''neighbourhood''' (or '''neighborhood''') is one of the basic concepts in a [[topological space]]. Intuitively speaking, a neighbourhood of a point is a [[Set (mathematics)|set]] containing the point where you can move that point some amount without leaving the set. | |||
This concept is closely related to the concepts of [[open set]] and [[interior (topology)|interior]]. | |||
== Definition == | |||
If <math>X</math> is a [[topological space]] and <math>p</math> is a point in <math>X</math>, a '''neighbourhood''' of <math>p</math> is a [[subset]] <math>V</math> of <math>X</math>, which includes an [[open set]] <math>U</math> containing <math>p</math>, | |||
:<math>p \in U \subseteq V.</math> | |||
This is also equivalent to <math>p\in X</math> being in the [[Interior (topology)#Interior_point|interior]] of <math>V</math>. | |||
Note that the neighbourhood <math>V</math> need not be an open set itself. If <math>V</math> is open it is called an '''open neighbourhood'''. Some [[scholar]]s require 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. | |||
The collection of all neighbourhoods of a point is called the [[neighbourhood system]] at the point. | |||
If <math>S</math> is a [[subset]] of <math>X</math> then a '''neighbourhood''' of <math>S</math> is a set <math>V</math> which includes an open set <math>U</math> containing <math>S</math>. It follows that a set <math>V</math> is a neighbourhood of <math>S</math> if and only if it is a neighbourhood of all the points in <math>S</math>. Furthermore, it follows that <math>V</math> is a neighbourhood of <math>S</math> [[iff]] <math>S</math> is a subset of the [[Interior (topology)|interior]] of <math>V</math>. | |||
== In a metric space == | |||
[[File:Neighborhood illust3.png|right|thumb|A set <math>S</math> in the plane and a uniform neighbourhood <math>V</math> of <math>S</math>.]] | |||
In a [[metric space]] <math>M = (X, d)</math>, a set <math>V</math> is a '''neighbourhood''' of a point <math>p</math> if there exists an [[open ball]] with centre <math>p</math> and radius <math>r>0</math>, such that | |||
:<math>B_r(p) = B(p;r) = \{ x \in X \mid d(x,p) < r \}</math> | |||
is contained in <math>V</math>. | |||
<math>V</math> is called '''uniform neighbourhood''' of a set <math>S</math> if there exists a positive number <math>r</math> such that for all elements <math>p</math> of <math>S</math>, | |||
:<math>B_r(p) = \{ x \in X \mid d(x,p) < r \}</math> | |||
is contained in <math>V</math>. | |||
For <math>r > 0</math> the '''<math>r</math>-neighbourhood''' <math>S_r</math> of a set <math>S</math> is the set of all points in <math>X</math> which are at distance less than <math>r</math> from <math>S</math> (or equivalently, <math>S</math><sub><math>r</math></sub> is the union of all the open balls of radius <math>r</math> which are centred at a point in <math>S</math>). | |||
It directly follows that an <math>r</math>-neighbourhood is a uniform neighbourhood, and that a set is a uniform neighbourhood if and only if it contains an <math>r</math>-neighbourhood for some value of <math>r</math>. | |||
== Examples == | |||
Given the set of [[real number]]s <math>\mathbb{R}</math> with the usual [[Euclidean metric]] and a subset <math>V</math> defined as | |||
:<math>V:=\bigcup_{n \in \mathbb{N}} B\left(n\,;\,1/n \right),</math> | |||
then <math>V</math> is a neighbourhood for the set <math>\mathbb{N}</math> of [[natural number]]s, but is ''not'' a uniform neighbourhood of this set. | |||
— | |||
[[Epsilon-neighborhood]] | |||
== 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 <math>X</math> is the assignment of a [[filter (mathematics)|filter]] <math>N(x)</math> (on the set <math>X</math>) to each <math>x</math> in <math>X</math>, such that | |||
# the point <math>x</math> is an element of each <math>U</math> in <math>N(x)</math> | |||
# each <math>U</math> in <math>N(x)</math> contains some <math>V</math> in <math>N(x)</math> such that for each <math>y</math> in <math>V</math>, <math>U</math> is in <math>N(y)</math>. | |||
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]] <math>S = (X, \delta)</math>, <math>V</math> is called a '''uniform neighbourhood''' of <math>P</math> if <math>P</math> is not [[closeness (mathematics)|close]] to <math>X \setminus V</math>, that is there exists no [[entourage (topology)|entourage]] containing <math>P</math> and <math>X \setminus V</math>. | |||
==Punctured neighbourhood== | |||
A '''punctured neighbourhood''' of a point <math>p</math> (sometimes called a '''deleted neighbourhood''') is a neighbourhood of <math>p</math>, without <math>\{p\}</math>. For instance, the [[interval (mathematics)|interval]] <math>(-1, 1) = \{y : -1 < y < 1\}</math> is a neighbourhood of <math>p = 0</math> in the [[real line]], so the set <math>(-1, 0) \cup (0, 1) = (-1, 1) \setminus \{0\}</math> is a punctured neighbourhood of <math>0</math>. Note that a punctured neighbourhood of a given point is not in fact a neighbourhood of the point. The concept of punctured neighbourhood occurs in the [[Limit_of_a_function#Functions_on_topological_spaces|definition of the limit of a function]]. | |||
==See also== | |||
* [[Tubular neighborhood]] | |||
==References== | |||
*{{cite book | |||
| last = Kelley | |||
| first = John L. | |||
| title = General topology | |||
| publisher = New York: Springer-Verlag | |||
| year = 1975 | |||
| pages = | |||
| isbn = 0-387-90125-6 | |||
}} | |||
*{{cite book | |||
| last = Bredon | |||
| first = Glen E. | |||
| authorlink = Glen Bredon | |||
| title = Topology and geometry | |||
| publisher = New York: Springer-Verlag | |||
| year = 1993 | |||
| pages = | |||
| isbn = 0-387-97926-3 | |||
}} | |||
*{{cite book | |||
| last = Kaplansky | |||
| first = Irving | |||
| authorlink = Irving Kaplansky | |||
| title = Set Theory and Metric Spaces | |||
| publisher = American Mathematical Society | |||
| year = 2001 | |||
| pages = | |||
| isbn = 0-8218-2694-8 | |||
}} | |||
[[Category:General topology]] | |||
[[Category:Mathematical analysis]] | |||
[[pt:Vizinhança]] |
Revision as of 09:28, 27 February 2013
28 year-old Painting Investments Worker Truman from Regina, usually spends time with pastimes for instance interior design, property developers in new launch ec Singapore and writing. Last month just traveled to City of the Renaissance.
In topology and related areas of mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a topological space. Intuitively speaking, a neighbourhood of a point is a set containing the point where you can move that point some amount without leaving the set.
This concept is closely related to the concepts of open set and interior.
Definition
If is a topological space and is a point in , a neighbourhood of is a subset of , which includes an open set containing ,
This is also equivalent to being in the interior of .
Note that the neighbourhood need not be an open set itself. If is open it is called an open neighbourhood. Some scholars require 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.
The collection of all neighbourhoods of a point is called the neighbourhood system at the point.
If is a subset of then a neighbourhood of is a set which 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, it follows that is a neighbourhood of iff is a subset of the interior of .
In a metric space
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 called uniform neighbourhood of a set if there exists a positive number such that for all elements of ,
For the -neighbourhood of a set is the set of all points in which are at distance less than from (or equivalently, is the union of all the open balls of radius which 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 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 (on the set ) to each in , such that
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 is not close to , that is there exists no entourage containing and .
Punctured neighbourhood
A punctured neighbourhood of a point (sometimes called a deleted neighbourhood) is a neighbourhood of , without . For instance, the interval is a neighbourhood of in the real line, so the set is a punctured neighbourhood of . Note that a punctured neighbourhood of a given point is not in fact a neighbourhood of the point. The concept of punctured neighbourhood occurs in the definition of the limit of a function.
See also
References
- 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534