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:B|UG}. 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

A compact metric space is second-countable. A separable metric space, e.g., a Polish space, is second-countable. (It is not true that separable first-countable spaces are second-countable; a counterexample is the real line equipped with the half-open or lower limit topology that has as basis the collection of half-open intervals [a,b)[a, b).)

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

A countable coproduct of second-countable spaces is second-countable.

If XX is second-countable and there is an open surjection f:XYf: X \to Y, then YY is second-countable.

Countable products of second-countable spaces are second-countable. Subspaces of second-countable spaces are second-countable.

For second-countable T_3 spaces X,YX, Y, if XX is locally compact, then the exponential Y XY^X is second-countable. Cf. Urysohn metrization theorem and Polish space. I (Todd Trimble) am uncertain to what extent the T 3T_3 assumption can be removed.

Revised on February 13, 2016 11:53:02 by Todd Trimble (50.154.86.95)