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In [[mathematics]], an '''axiom of countability''' is a property of certain [[mathematical object]]s that asserts the existence of a [[countable|countable set]] with certain properties. Without such an axiom, such a set might not provably exist.

==Important examples==
Important countability axioms for [[topological space]]s include:<ref>{{citation|title=Modern General Topology|series=North-Holland Mathematical Library|first=J.-I.|last=Nagata|edition=3rd|publisher=Elsevier|year=1985|isbn=9780080933795|page=104|url=https://books.google.com/books?id=ecvd8dCAQp0C&pg=PA104}}.</ref>
*[[sequential space]]: a set is open if every [[sequence]] [[limit of a sequence|convergent]] to a [[point (geometry)|point]] in the set is eventually in the set
*[[first-countable space]]: every point has a countable [[neighbourhood system|neighbourhood basis]] (local base)
*[[second-countable space]]: the topology has a countable [[base (topology)|base]]
*[[separable space]]: there exists a countable [[dense (topology)|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.
*Every [[metric space]] is first countable.
*For metric spaces, second-countability, separability, and the Lindelöf property are all equivalent.

==Related concepts==
Other examples of mathematical objects obeying axioms of countability include [[sigma-finite]] [[measure (mathematics)|measure space]]s, and [[lattice (order)|lattice]]s of [[countable type]].

==References==
{{reflist}}
{{sia|mathematics}}
[[Category:General topology]]
[[Category:Mathematical axioms]]

Axiom of countability - Revision history

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