Concept

Axiom of countability

Résumé
In mathematics, an axiom of countability is a property of certain mathematical objects that asserts the existence of a countable set with certain properties. Without such an axiom, such a set might not provably exist. Important examples Important countability axioms for topological spaces include: *sequential space: a set is open if every sequence convergent to a point in the set is eventually in the set *first-countable space: every point has a countable neighbourhood basis (local base) *second-countable space: the topology has a countable base *separable space: there exists a countable dense subset *Lindelöf space: every open cover has a countable subcover *σ-compact space: there exists a countable cover by compact spaces Relationships with each other These axioms are related to each other in the following ways: *Every first-countable space is sequential. *Every second-countable space is first countable, separable, and Lindelöf. *Every σ-compact space is Lindelöf
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