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In
topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
and related areas of mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
. It is closely related to the concepts of
open set In mathematics, open sets are a generalization of open intervals in the real line. In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are ...
and
interior Interior may refer to: Arts and media * ''Interior'' (Degas) (also known as ''The Rape''), painting by Edgar Degas * ''Interior'' (play), 1895 play by Belgian playwright Maurice Maeterlinck * ''The Interior'' (novel), by Lisa See * Interior de ...
. 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

If X is a
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
and p is a point in X, then a of p is a
subset In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset o ...
V of X that includes an
open set In mathematics, open sets are a generalization of open intervals in the real line. In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are ...
U containing p, p \in U \subseteq V \subseteq X. This is also equivalent to the point p \in X belonging to the topological interior of V in X. The neighbourhood V need be an open subset X, but when V is open in X then it is called an . Some authors have been known to require neighbourhoods to 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 In topology and related areas of mathematics, the neighbourhood system, complete system of neighbourhoods, or neighbourhood filter \mathcal(x) for a point x in a topological space is the collection of all neighbourhoods of x. Definitions Neighbour ...
at the point.


Neighbourhood of a set

If S is a
subset In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset o ...
of a topological space X, then a neighbourhood of S is a set V that includes an open set U containing S,S \subseteq U \subseteq V \subseteq X.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 In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bic ...
S is a subset of the
interior Interior may refer to: Arts and media * ''Interior'' (Degas) (also known as ''The Rape''), painting by Edgar Degas * ''Interior'' (play), 1895 play by Belgian playwright Maurice Maeterlinck * ''The Interior'' (novel), by Lisa See * Interior de ...
of V. A neighbourhood of S that is also an open subset of X is called an of S. The neighbourhood of a point is just a special case of this definition.


In a metric space

In a
metric space In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general set ...
M = (X, d), a set V is a neighbourhood of a point p if there exists an
open ball In mathematics, a ball is the solid figure bounded by a ''sphere''; it is also called a solid sphere. It may be a closed ball (including the boundary points that constitute the sphere) or an open ball (excluding them). These concepts are defin ...
with center p and radius r>0, such that B_r(p) = B(p; r) = \ is contained in V. V is called uniform neighbourhood of a set S if there exists a positive number r such that for all elements p of S, B_r(p) = \ is contained in V. For r > 0, the r-neighbourhood S_r of a set S is the set of all points in X that are at distance less than r from S (or equivalently, S_r is the union of all the open balls of radius r that are centered at a point in S): S_r = \bigcup\limits_ B_r(p). It directly follows that an r-neighbourhood is a uniform neighbourhood, and that a set is a uniform neighbourhood if and only if it contains an r-neighbourhood for some value of r.


Examples

Given the set of
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one- dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Ever ...
s \R with the usual
Euclidean metric In mathematics, the Euclidean distance between two points in Euclidean space is the length of a line segment between the two points. It can be calculated from the Cartesian coordinates of the points using the Pythagorean theorem, therefore o ...
and a subset V defined as V := \bigcup_ B\left(n\,;\,1/n \right), then V is a neighbourhood for the set \N of
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''cardinal ...
s, but is a uniform neighbourhood of this set.


Topology from neighbourhoods

The above definition is useful if the notion of
open set In mathematics, open sets are a generalization of open intervals in the real line. In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are ...
is already defined. There is an alternative way to define a topology, by first defining the
neighbourhood system In topology and related areas of mathematics, the neighbourhood system, complete system of neighbourhoods, or neighbourhood filter \mathcal(x) for a point x in a topological space is the collection of all neighbourhoods of x. Definitions Neighbour ...
, and then open sets as those sets containing a neighbourhood of each of their points. A neighbourhood system on X is the assignment of a
filter Filter, filtering or filters may refer to: Science and technology Computing * Filter (higher-order function), in functional programming * Filter (software), a computer program to process a data stream * Filter (video), a software component that ...
N(x) of subsets of X to each x in X, such that # the point x is an element of each U in N(x) # each U in N(x) contains some V in N(x) such that for each y in V, U is in N(y). 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 neighbourhoods

In a
uniform space In the mathematical field of topology, a uniform space is a set with a uniform structure. Uniform spaces are topological spaces with additional structure that is used to define uniform properties such as completeness, uniform continuity and un ...
S = (X, \Phi), V is called a uniform neighbourhood of P if there exists an entourage U \in \Phi such that V contains all points of X that are U-close to some point of P; that is, U \subseteq V for all x \in P.


Deleted neighbourhood

A deleted neighbourhood of a point p (sometimes called a punctured neighbourhood) is a neighbourhood of p, without \. For instance, the interval (-1, 1) = \ is a neighbourhood of p = 0 in the
real line In elementary mathematics, a number line is a picture of a graduated straight line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real number to a poin ...
, so the set (-1, 0) \cup (0, 1) = (-1, 1) \setminus \ is a deleted neighbourhood of 0. 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

* * *


References

* * *{{cite book , last = Kaplansky , first = Irving , author-link = Irving Kaplansky , year = 2001 , title = Set Theory and Metric Spaces , publisher = American Mathematical Society , isbn = 0-8218-2694-8 General topology Mathematical analysis