# nLab second-countable space

### Context

#### Topology

topology

algebraic topology

# 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 $B$.

###### Definition

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

$G=\bigvee \left\{U:B\phantom{\rule{thickmathspace}{0ex}}\mid \phantom{\rule{thickmathspace}{0ex}}U\subseteq G\right\}.$G = \bigvee \{ U\colon B \;|\; U \subseteq G \} .

## 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.

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