• Source: Axiom of countability

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.
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 spaces, and lattices of countable type.




References

Kata Kunci Pencarian:

• Axiom of countability
• Axiom of countable choice
• Second-countable space
• First-countable space
• List of axioms
• Countable set
• Axiom of choice
• Dedekind-infinite set
• Axiom of dependent choice
• Aleph number