PlanetMath: second countable

(more info)

Math for the people, by the people.

donor list - find out how

Main Menu

second countable

(Definition)

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) ]

(view preamble | get metadata)

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

Log in to rate this entry.
(view current ratings)

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 14688 times total.

Classification:

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

Pending Errata and Addenda

None.[ View all 4 ]

Discussion

forum policy

No messages.

Interact