nLab
second-countable space

Second-countable spaces

Idea

A space (such as a topological space) is second-countable if, in a certain sense, there is only a countable amount of information globally in its topology. (Change ‘globally’ to ‘locally’ to get a first-countable space.)

Definitions

Definition

A topological space is second-countable if it has a countable basis BB.

Definition

A locale is second-countable if there is a countable set BB of open subspaces (elements of the frame of opens) such that every open GG is a join of some subset of BB. That is, we have

G={U:BUG}. G = \bigvee \{ U\colon B \;|\; U \subseteq G \} .

Generalisations

The weight of a space is the minimum of the cardinalities of the possible bases BB. We are implicitly using the axiom of choice here, to suppose that this set of cardinalities (which really is a small set because bounded above by the number of open subspaces, and inhabited by this number as well) has a minimum. But without Choice, we can still consider this collection of cardinalities.

Then a second-countable space is simply one with a countable weight.

Examples

Any space first-countable space must be second-countable. (Conversely, any second-countable space must be first-countable.) In particular, separable metric spaces are second-countable.

A topological manifold is second-countable iff it is paracompact and has countably many connected components.

Revised on September 8, 2012 08:29:17 by Toby Bartels (98.23.143.147)