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 {Un} such that every open set V of X can be written as the union of some of the Un. (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.)

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