Neighbourhood (topology)

In topology, a neighbourhood of a point x in a topological space X is a set N such that x is in the interior of N; that is, there is an open set U such that $$x \in U \subseteq N$$. A neighbourhood of a set A in X is a set N such that A is contained in the interior of N; that is, there is an open set U such that $$A \subseteq U \subseteq N$$.

A topology may be defined in terms of its neighbourhood structure: a set is open if and only if it is a neighbourhood of each of its points.