nLab
first-countable space

First-countable spaces

Idea

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

Definitions

Definition

A topological space is first-countable if every point xx has a countable local basis B xB_x.

Generalisations

The character of a space at a point xx is the minimum of the cardinalities of the possible bases B xB_x. 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 neighbourhoods of xx, and inhabited by this number as well) has a minimum. But without Choice, we can still consider this collection of cardinalities.

Then a first-countable space is simply one whose characters are all countable.

The character, tout court, of a space is the supremum of the characters of its points; then a first-countable space is simply one with a countable character.

Examples

Any second-countable space must also be first-countable.

Any metric space is first-countable.

Created on September 8, 2012 08:35:00 by Toby Bartels (98.23.143.147)