PlanetMath: second countable

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second countable

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A topological space is said to be second countable if it has a countable basis. It can be shown that a second countable space is both Lindelöf and separable, although the converses fail. For instance, the lower limit topology on the real line is both Lindelöf and separable, but not second countable.

"second countable" is owned by rspuzio. [ full author list (2) | owner history (1) ]

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Other names:

second axiom of countability, completely separable, perfectly separable, second-countable

Attachments:: every second countable space is separable (Proof) by drini
second countability is hereditary (Theorem) by matte
a compact metric space is second countable (Theorem) by azdbacks4234

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Cross-references: line, real, lower limit topology, converses, separable, Lindelöf, countable, topological space
There are 22 references to this entry.

This is version 13 of second countable, born on 2002-01-01, modified 2008-09-12.
Object id is 1162, canonical name is SecondCountable.
Accessed 14809 times total.

Classification:

54D70 (General topology :: Fairly general properties :: Base properties)

Pending Errata and Addenda

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