Neighbourhood (topology)

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 (2-3) 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 for x 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 set of B(x) contains an open neighbourhood V of x,
   i.e., a set V (not necessarily a member of B(x)) that contains some set of B(y) for all of its elements y in V.

Axiom (2) implies that B(x) is a filter base.
The family N(x) consisting of all sets containing a set of B(x) is the neighbourhood filter generated by B(x)
If the family B(x) is countable for all x in X, then the corresponding topological (or, equivalently, neighbourhood) space is said to be first-countable

Example: Metric spaces

In a metric space the (open or closed) balls with centre x are a neighbourhood base for x and define the topology induced by the metric.
Moreover, it is sufficient to take the balls with radius 1/n for all natural numbers n (that is, a countable set for each point x, therefore metric spaces are first-countable).

The classical example (in calculus or real analysis) is $\mathbb {R} ^{d}$:

A neighbourhood base for $x\in \mathbb {R} ^{d}$ is given by

${\mathcal {B}}(x)=\left\{B(x,n)\mid n\in \mathbb {N} \right\}$

where

$B(x,n)=\left\{y\left\vert \left|y-x\right|<{1 \over n}\right.,y\in \mathbb {R} ^{d}\right\}\subset \mathbb {R} ^{d}$

are the open balls with centre $x$ and radius $1/n$.

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 neighbourhood base for this point.)