Neighbourhood (mathematics) - in A Metric Space

In A Metric Space

In a metric space, a set is a neighbourhood of a point if there exists an open ball with centre and radius, such that

is contained in .

is called uniform neighbourhood of a set if there exists a positive number such that for all elements of ,

is contained in .

For the -neighbourhood of a set is the set of all points in which are at distance less than from (or equivalently, is the union of all the open balls of radius which are centred at a point in ).

It directly follows that an -neighbourhood is a uniform neighbourhood, and that a set is a uniform neighbourhood if and only if it contains an -neighbourhood for some value of .