secondcountable

Second-countable is a property of topological spaces. A space X is second-countable if its topology has a countable base: there exists a countable collection B of open sets such that every open set in X can be written as a union of elements of B.

Key consequences and relationships: If X is second-countable, then X is separable; a countable dense subset can be obtained by selecting a point from each nonempty member of a base. Second-countable spaces are also Lindelöf, meaning every open cover has a countable subcover. Subspaces of a second-countable space are second-countable, since restricting the base to the subspace yields a countable base as well. The product of countably many second-countable spaces is second-countable, but the product of uncountably many such spaces need not be. In metric spaces, second-countability is equivalent to separability: a metric space is second-countable if and only if it is separable. If a space is regular and second-countable, it is metrizable (Urysohn metrization theorem).

Examples and non-examples: The real line with its standard topology is second-countable; any subspace of R^n is second-countable. A countable discrete space is second-countable, while an uncountable discrete space is not. An uncountable product of spaces with the product topology is typically not second-countable.

Related concepts include first-countable, separable, Lindelöf, metrizable, and base in topology. Second-countability is a central condition in many classical theorems, especially in analysis and geometry, where it often ensures manageable cardinality and favorable covering properties.