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 countability axioms for topological spaces include: [1]
- 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
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.
Other examples of mathematical objects obeying axioms of countability include sigma-finite measure spaces, and lattices of countable type.
- ^ Nagata, J.-I. (1985), Modern General Topology, North-Holland Mathematical Library (3rd ed.), Elsevier, p. 104, ISBN 9780080933795.
If an internal link incorrectly led you here, you may wish to change the link to point directly to the intended article.
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 countability axioms for topological spaces include: [1]
- 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
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.
Other examples of mathematical objects obeying axioms of countability include sigma-finite measure spaces, and lattices of countable type.
- ^ Nagata, J.-I. (1985), Modern General Topology, North-Holland Mathematical Library (3rd ed.), Elsevier, p. 104, ISBN 9780080933795.
If an internal link incorrectly led you here, you may wish to change the link to point directly to the intended article.