# nLab second-countable space

Second-countable spaces

# 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

(second-countable topological space)

A topological space is second-countable if it has a base for its topology consisting of a countable set of subsets.

###### 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

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

## 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

###### Example

(Euclidean space is second-countable)

Let $n \in \mathbb{N}$. Consider the Euclidean space $\mathbb{R}^n$ with its Euclidean metric topology. Then $\mathbb{R}^n$ is second countable.

A countable set of base open subsets is given by the open balls $B^\circ_x(\epsilon)$ of rational radius $\epsilon \in \mathbb{Q}_{\geq 0} \subset \mathbb{R}_{\geq 0}$ and centered at points with rational coordinates: $x \in \mathbb{Q}^n \subset \mathbb{R}^n$.

###### Example

A compact metric space is second-countable.

###### Example

A separable metric space, e.g., a Polish space, is second-countable.

###### Remark

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 $[a, b)$.

###### Example

A Hausdorff locally Euclidean space is second-countable precisely it is paracompact and has a countable set of connected components. In this case it is called a topological manifold.

See at topological space this prop..

###### Example

A countable coproduct (disjoint union space) of second-countable spaces is second-countable.

Countable products (product topological spaces) of second-countable spaces are second-countable.

Subspaces of second-countable spaces are second-countable.

###### Example

If $X$ is second-countable and there is an open surjection $f \colon X \to Y$, then $Y$ is second-countable.

###### Example

For second-countable T_3 spaces $X, Y$, if $X$ is locally compact, then the mapping space $Y^X$ with the compact-open topology is second-countable.

Cf. Urysohn metrization theorem and Polish space. I (Todd Trimble) am uncertain to what extent the $T_3$ assumption can be removed.

## Properties

propertiesimplications
metrisable: topology is induced by a metrica metric space has a $\sigma$-locally discrete base as a corollary of the fact that open covers of metric spaces have open $\sigma$-locally discrete refinements: take $\sigma$-locally discrete refinements of the covers by $1/n$-balls for $n=1,2,\ldots$.
second-countable: there is a countable base of the topology.Nagata-Smirnov metrization theorem
$\sigma$-locally discrete base, i.e. $X$ has a $\sigma$-locally discrete base.a second-countable space has a $\sigma$-locally finite base: take the the collection of singeltons of all elements of countable cover of $X$.
$\sigma$-locally finite base, i.e. $X$ has a countably locally finite base.second-countable spaces are separable: choose a point in each set of countable cover.
separable: there is a countable dense subset.second-countable spaces are Lindelöf
Lindelöf: every open cover has a countable sub-cover.weakly Lindelöf spaces with countably locally finite base are second countable
weakly Lindelöf: every open cover has a countable subcollection the union of which is dense.separable metacompact spaces are Lindelöf
countable choice: the natural numbers is a projective object in Set.separable spaces are weakly Lindelöf: given a countable dense subset and an open cover choose for each point of the subset an open from the cover.
metacompact: every open cover has a point-finite open refinement.Lindelöf spaces are trivially also weakly Lindelöf.
first-countable: every point has a countable neighborhood basea space with a $\sigma$-locally finite base is first countable: obviously, every point is contained in at most countably many sets of a $\sigma$-locally finite base.
Frechet-Uryson space: the closure of a set $A$ consists precisely of all limit points of sequences in $A$a first-countable space is Fréchet-Urysohn: obvious
sequential topological space: a set $A$ is closed if it contains all limit points of sequences in $A$obviously, a Fréchet-Uryson space is sequential
countably tight: for each subset $A$ and each point $x\in \overline A$ there is a countable subset $D\subseteq A$ such that $x\in \overline D$.obviously, a sequential space is countably tight.

Last revised on April 5, 2019 at 19:51:00. See the history of this page for a list of all contributions to it.