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

Generalisations

The weight of a space is the minimum of the cardinalities of the possible bases $B$. 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.

Related concepts

• first-countable space
• second-countable space
• separable space