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 X {displaystyle X} is a topological space and p {displaystyle p} is a point in X {displaystyle X} , a neighbourhood of p {displaystyle p} is a subset V {displaystyle V} of X {displaystyle X} that includes an open set U {displaystyle U} containing p {displaystyle p} , This is also equivalent to p ∈ X {displaystyle pin X} being in the interior of V {displaystyle V} . Note that the neighbourhood V {displaystyle V} need not be an open set itself. If V {displaystyle V} is open it is called an open neighbourhood. Some mathematicians require that neighbourhoods be open, so it is important to note conventions.