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Axiom of countability
From Wikipedia, the free encyclopedia
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 spaces: a set is open if every sequence converging to a point in the set is eventually in the set,
- first-countable spaces: every point has a countable local base,
- second-countable spaces: the topology has a countable base,
- separable spaces: there exists a countable dense subspace,
- Lindelöf spaces: every open cover has a countable subcover,
- σ-compact spaces: there exists a countable cover by compact spaces,
These axioms are not all unrelated. In particular, every second-countable space is first-countable, separable, and Lindelöf. Also, every σ-compact space is Lindelöf. For metric spaces, first-countability is automatic, and second-countability, separability, and the Lindelöf property are all equivalent.
Other examples:
- sigma-finite measure spaces
- lattices of countable type