Second-Countable Space

(General Topology)

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General Topology
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What is 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. Many "well-behaved" spaces in mathematics are second-countable. For example, Euclidean space (Rn) with its usual topology is second-countable. Although the usual base of open balls is uncountable, one can restrict to the collection of all open balls with rational radii and whose centers have rational coordinates. This restricted set is countable and still forms a basis.

Technology Types

general topologypossessionproperties of topological space

Synonyms

Completely separable spaceSecond countabilitySecond countability axiomSecond countableSecond countable topologySecond countablility axiomSecond-countabilitySecond-countable

Translations

Espace à base dénombrable (fr)Segon axioma de numerabilitat (ca)Segundo axioma de numerabilidad (es)Вторая аксиома счётности (ru)Друга аксіома зліченності (uk)제2 가산 공간 (ko)第二可算的空間 (ja)

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