nLab second-countable space

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

• second-countable regular spaces are paracompact

• locally compact and second-countable spaces are sigma-compact

Related countability properties

Axioms: axiom of choice (AC), countable choice (CC).

Properties

• second-countable: there is a countable base of the topology.

• metrisable: the topology is induced by a metric.

• $\sigma$-locally discrete base: the topology of $X$ is generated by a $\sigma$-locally discrete base.

• $\sigma$-locally finite base: the topology of $X$ is generated by a countably locally finite base.

• separable: there is a countable dense subset.

• Lindelöf: every open cover has a countable sub-cover.

• weakly Lindelöf: every open cover has a countable subcollection the union of which is dense.

• metacompact: every open cover has a point-finite open refinement.

• countable chain condition: A family of pairwise disjoint open subsets is at most countable.

• first-countable: every point has a countable neighborhood base

• Frechet-Uryson space: the closure of a set $A$ consists precisely of all limit points of sequences in $A$

• sequential topological space: a set $A$ is closed if it contains all limit points of sequences in $A$

• countably tight: for each subset $A$ and each point $x\in \overline A$ there is a countable subset $D\subseteq A$ such that $x\in \overline D$.

Implications

• a metric space has a $\sigma$-locally discrete base

• Nagata-Smirnov metrization theorem

• a second-countable space has a $\sigma$-locally finite base: take the the collection of singeltons of all elements of a countable cover of $X$.

• second-countable spaces are separable: use the axiom of countable choice to choose a point in each set of a countable cover.

• second-countable spaces are Lindelöf.

• weakly Lindelöf spaces with countably locally finite base are second countable.

• separable metacompact spaces are Lindelöf.

• separable spaces satisfy the countable chain condition: given a dense set $D$ and a family $\{U_\alpha : \alpha \in A\}$, the map $D \cap \bigcup_{\alpha \in A} U_\alpha \to A$ assigning $d$ to the unique $\alpha \in A$ with $d \in U_\alpha$ is surjective.

• 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.

• Lindelöf spaces are trivially also weakly Lindelöf.

• a 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.

• a first-countable space is obviously Fréchet-Urysohn.

• a Fréchet-Uryson space is obviously sequential.

• a sequential space is obviously countably tight.

• paracompact spaces satisfying the countable chain condition are Lindelöf.

References

• Felix Cherubini, Thierry Coquand, Freek Geerligs, Hugo Moeneclaey, A Foundation for Synthetic Stone Duality (arXiv:2412.03203)