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]

See also

• neighbourhood

References

1. ^ ^{a b} Mendelson, Bert (1975). Introduction to Topology. New York: Dover. pp. 77. ISBN 978-0-486-66352-4.