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In 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:

- sequential space: a set is open if every sequence converging to a point in the set is eventually in the set (at least from a certain term)
- 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 subspace
- Lindelöf space: every open cover has a countable subcover
- σ-compact space: there exists a countable cover by compact spaces

Relations:

- Every first countable space is sequential.
- Every second-countable space is first-countable, separable, and Lindelöf.
- Every σ-compact space is Lindelöf.
- A metric space is first-countable.
- For metric spaces second-countability, separability, and the Lindelöf property are all equivalent.

Other examples:

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Last updated on Saturday August 30, 2008 at 08:40:08 PDT (GMT -0700)

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This article is licensed under the GNU Free Documentation License.

Last updated on Saturday August 30, 2008 at 08:40:08 PDT (GMT -0700)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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