Neighbourhood space

In topology and related areas of mathematics, a neighbourhood space is a set X such that for each $x\in X$ there is an associated neighbourhood system ${\mathfrak{\Re }}_x$.

A subset O of a neighbourhood space is called open if for every $x\in O$ is a neighbourhood of x. Under this definition the open sets of a neighbourhood space give rise to a topological space. Conversely, every topological space is a neighbourhood space under the usual definition of a neighbourhood in a topological space.

See also

• neighbourhood