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 "wiggle" or "move" the point a bit without leaving the set. Topology (Greek topos, place and logos, study) is a branch of mathematics concerned with the study of topological spaces. ... Mathematics, often abbreviated maths in Commonwealth English and math in American English, is the study of abstraction. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... This article is about sets in mathematics. ...

This concept is closely related to the concepts of open set and interior. In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U... In mathematics, the interior of a set S consists of all points which are intuitively not on the edge of S. A point which is in the interior of S is an interior point of S. The notion of interior is in many ways dual to the notion of closure. ...

Definition

For an arbitrary topological space X, V is called neighbourhood of a point p (or a subset S), if p is an element of (or S is contained in) some open set contained in V. Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U...

The collection of all neighbourhoods for a point is called the neighbourhood system for the point. In topology and related areas of mathematics, the neighbourhood system or neighbourhood filter for a point x is the collection of all neighbourhoods for the point x. ...

The case of a metric space

In a metric space M = (X,d), a set V is a neighbourhood for a point p if there exists an open ball with center p and radius r, In mathematics, a metric space is a set (or space) where a distance between points is defined. ... A synonym for ball (in geometry or topology, and in any dimension) is disk (or disc Geometry In metric geometry, a ball is a set containing all points within a specified distance of a given point. ...

which is contained in V.

A set V is a neighbourhood for a set S if V is a neighbourhood for all elements of S.

V is called uniform neighbourhood for a set S if there exists a positive number r such that for all elements p of S,

is contained in V.

Examples

Given the set of real numbers R with the usual Euclidean metric and a subset V defined as Please refer to Real vs. ... The Euclidean distance of two points x = (x1,...,xn) and y = (y1,...,yn) in Euclidean n-space is computed as It is the ordinary distance between the two points that one would measure with a ruler, which can be proven by repeated application of the Pythagorean theorem. ...

then V is a neighbourhood for the set N of natural numbers, but is not a uniform neighbourhood of this set. Natural number can mean either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). Natural numbers have two main purposes: they can be used for counting (there are 3 apples on the table), or they can be used for ordering (this is...

Topology from neigborhoods

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. In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U... In topology and related areas of mathematics, the neighbourhood system or neighbourhood filter for a point x is the collection of all neighbourhoods for the point x. ...

A neighborhood system on X is the assignment of a filter N(x) (on the set X) to each x in X, such that The term filter may refer to: Filter (chemistry) — a device to separate mixtures, e. ...

1. the point x is an element of each U in N(x)
2. 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, 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.

Generalized definition

In a uniform space S:=(X, δ) V is called a uniform neighbourhood of P if P is not close to X V, that is there exists no entourage containing P and X V. In topology, one defines uniform spaces in order to study concepts such as uniform continuity, completeness and uniform convergence. ... In topology and related areas in mathematics closeness is one of the basic concepts in a topological space. ... In topology, one defines uniform spaces in order to study concepts such as uniform continuity, completeness and uniform convergence. ...