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.)