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.
A set in the plane is a neighbourhood of a point if a small disc around is contained in .
This is also equivalent to being in the interior of .
The neighbourhood need not be an open set itself. If is open it is called an open neighbourhood. Some mathematicians require neighbourhoods to be open, so it is important to note conventions.
A closed rectangle is not a neighbourhood of any of its corners (or points on the boundary).
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.
If S is a subset 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 Sif and only ifS is a subset of the interior of V.
A neighbourhood of S that is also an open set is called an open neighbourhood of S.
The neighbourhood of a point is just a special case of this definition.
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, 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.
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.
^Dixmier, Jacques (1984). General Topology. Undergraduate Texts in Mathematics. Translated by Sterling K. Berberian. Springer. p. 6. ISBN0-387-90972-9. According to this definition, an open neighborhood of x is nothing more than an open subset of E that contains x.
Kelley, John L. (1975). General topology. New York: Springer-Verlag. ISBN0-387-90125-6.