TheInfoList

In
topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

and related areas of
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, a neighbourhood (or neighborhood) is one of the basic concepts in a
topological space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...
. It is closely related to the concepts of
open set In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
and
interior Interior may refer to: Arts and media * Interior (Degas), ''Interior'' (Degas) (also known as ''The Rape''), painting by Edgar Degas * Interior (play), ''Interior'' (play), 1895 play by Belgian playwright Maurice Maeterlinck * The Interior (novel) ...
. 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 Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...
and $p$ is a point in $X,$ a of $p$ is a
subset In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

$V$ of $X$ that includes an
open set In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
$U$ containing $p,$ $p \in U \subseteq V.$ This is also equivalent to the point $p \in X$ belonging to the
topological interior In mathematics, specifically in general topology, topology, the interior of a subset of a topological space is the Union (set theory), union of all subsets of that are Open set, open in . A point that is in the interior of is an interior point ...

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 at the point.

Neighbourhood of a set

If $S$ is a
subset In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

of topological space $X$ then a neighbourhood of $S$ is a set $V$ that includes an open set $U$ containing $S.$ 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 Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fallacies. Formal logic represents st ...
$S$ is a subset of the
interior Interior may refer to: Arts and media * Interior (Degas), ''Interior'' (Degas) (also known as ''The Rape''), painting by Edgar Degas * Interior (play), ''Interior'' (play), 1895 play by Belgian playwright Maurice Maeterlinck * The Interior (novel) ...
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 Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gene ...
$M = \left(X, d\right),$ a set $V$ is a neighbourhood of a point $p$ if there exists an
open ball In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
with centre $p$ 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 $S$): $S_r = \bigcup\limits_ B_r\left(p\right).$ 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 Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
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 occas ...
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 total order, ordering (as in "this is the ''third'' largest city in the country"). In common mathematical terminology, w ...
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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
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 $X$ is the assignment of a
filter Filter, filtering or filters may refer to: Science and technology Device * Filter (chemistry), a device which separates solids from fluids (liquids or gases) by adding a medium through which only the fluid can pass ** Filter (aquarium), critical ...
$N\left(x\right)$ of subsets of $X$ to each $x$ in $X,$ such that # the point $x$ is an element of each $U$ in $N\left(x\right)$ # each $U$ in $N\left(x\right)$ contains some $V$ in $N\left(x\right)$ such that for each $y$ in $V,$ $U$ is in $N\left(y\right).$ 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 s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathemati ...
$S = \left(X, \Phi\right),$ $V$ 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, 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 $\left(-1, 1\right) = \$ is a neighbourhood of $p = 0$ in the
real line In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
, so the set $\left(-1, 0\right) \cup \left(0, 1\right) = \left(-1, 1\right) \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).