Neighbourhood (mathematics)

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. If is a topological space and is a point in then a of is a subset of that includes an open set containing , This is also equivalent to the point belonging to the topological interior of in The neighbourhood need be an open subset of but when is open in 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 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. 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 of The neighbourhood of a point is just a special case of this definition.