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 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 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.

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First-countable space

In topology, a branch of mathematics, a first-countable space is a topological space satisfying the "first axiom of countability". Specifically, a space is said to be first-countable if each point has a countable neighbourhood basis (local base). That is, for each point in there exists a sequence of neighbourhoods of such that for any neighbourhood of there exists an integer with contained in Since every neighborhood of any point contains an open neighborhood of that point, the neighbourhood basis can be chosen without loss of generality to consist of open neighborhoods.

Second-countable space

In topology, a second-countable space, also called a completely separable space, is a topological space whose topology has a countable base. More explicitly, a topological space is second-countable if there exists some countable collection of open subsets of such that any open subset of can be written as a union of elements of some subfamily of . A second-countable space is said to satisfy the second axiom of countability. Like other countability axioms, the property of being second-countable restricts the number of open sets that a space can have.

Axiom of countability

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