# 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 $\mathfrak{R}_x$.[1]
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.[1]