In topology and related areas of mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a topological space. It is closely related to the concepts of open set and interior. 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.
![](http://upload.wikimedia.org/wikipedia/commons/thumb/7/79/Neighborhood_illust1.svg/220px-Neighborhood_illust1.svg.png)
Definitions
Neighbourhood of a point
If is a topological space and
is a point in
then a neighbourhood[1] of
is a subset
of
that includes an open set
containing
,
This is equivalent to the point belonging to the topological interior of
in
The neighbourhood need not be an open subset of
When
is open (resp. closed, compact, etc.) in
it is called an open neighbourhood[2] (resp. closed neighbourhood, compact neighbourhood, etc.). Some authors[3] require neighbourhoods to be open, so it is important to note their conventions.
![](http://upload.wikimedia.org/wikipedia/commons/thumb/2/21/Neighborhood_illust2.svg/220px-Neighborhood_illust2.svg.png)
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 closed 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 is a subset of a topological space
, then a neighbourhood of
is a set
that 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,
is a neighbourhood of
if and only if
is a subset of the interior of
A neighbourhood of
that is also an open subset of
is called an open neighbourhood of
The neighbourhood of a point is just a special case of this definition.
In a metric space
![](http://upload.wikimedia.org/wikipedia/commons/thumb/d/d0/Neighborhood_illust3.svg/220px-Neighborhood_illust3.svg.png)
![](http://upload.wikimedia.org/wikipedia/commons/thumb/c/c1/Epsilon_Umgebung.svg/220px-Epsilon_Umgebung.svg.png)
In a metric space a set
is a neighbourhood of a point
if there exists an open ball with center
and radius
such that
is contained in
is called a uniform neighbourhood of a set
if there exists a positive number
such that for all elements
of
is contained in
Under the same condition, for the
-neighbourhood
of a set
is the set of all points in
that are at distance less than
from
(or equivalently,
is the union of all the open balls of radius
that are centered 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
![](http://upload.wikimedia.org/wikipedia/commons/thumb/1/1a/Set_of_real_numbers_with_epsilon-neighbourhood.svg/220px-Set_of_real_numbers_with_epsilon-neighbourhood.svg.png)
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
of subsets of
to each
in
such that
- the point
is an element of each
in
- each
in
contains some
in
such that for each
in
is in
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
is called a uniform neighbourhood of
if there exists an entourage
such that
contains all points of
that are
-close to some point of
that is,
for all
Deleted neighbourhood
A deleted neighbourhood of a point (sometimes called a punctured neighbourhood) is a neighbourhood of
without
For instance, the interval
is a neighbourhood of
in the real line, so the set
is a deleted neighbourhood of
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).[4]
See also
- Isolated point – Point of a subset S around which there are no other points of S
- Neighbourhood system – (for a point x) collection of all neighborhoods for the point x
- Region (mathematics) – Connected open subset of a topological space
- Tubular neighbourhood – neighborhood of a submanifold homeomorphic to that submanifold’s normal bundle
Notes
References
- Bredon, Glen E. (1993). Topology and geometry. New York: Springer-Verlag. ISBN 0-387-97926-3.
- Engelking, Ryszard (1989). General Topology. Heldermann Verlag, Berlin. ISBN 3-88538-006-4.
- Kaplansky, Irving (2001). Set Theory and Metric Spaces. American Mathematical Society. ISBN 0-8218-2694-8.
- Kelley, John L. (1975). General topology. New York: Springer-Verlag. ISBN 0-387-90125-6.
- Willard, Stephen (2004) [1970]. General Topology. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-43479-7. OCLC 115240.