In

general topology, a

topological space X is said to be

*second countable*, or satisfy the

*second axiom of countability*, if there is a

countable family of

open sets {U

_{n}} such that every

open set V of X can be written as the

union of some of the U

_{n}. (Such a family in general is called a

base or

basis for the

topology of X, but this usage is rare outside the study of

topology per se, so I have not

noded it.) A

second countable space is

separable, and for

metric spaces the

converse holds: a

separable metric space is

second countable. (Not for

spaces which are not

metrizable!)

Generally, if a space is second countable and also satisfies some separation axiom (for instance if it is regular or normal), then there are few if any surprises in its topology; it is likely to be metrizable and possibly even a topological manifold, depending on the context. (The Urysohn metrization theorem says that a regular, second countable space is metrizable.)