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Axiom Of CountabilityIn mathematics, an axiom of countability is a property of certain mathematical objects (usually in a category) that requires the existence of a countable set with certain properties, while without it such sets might not exist. Important countability axioms for topological spaces: These axioms are not all unrelated. In particular, every second-countable space is first-countable, separable, and Lindelf. Also, every σ-compact space is Lindelf. For metric spaces, first-countability is automatic, and second-countability, separability, and the Lindelf property are all equivalent. Other examples:
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