First and second countable topological space

• Idea
• Definitions
• Generalisations
• Examples
• Properties
• Related countability properties
• Properties
• Implications

(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.

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

(Euclidean space is second-countable)

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

A countable set of base open subsets is given by the open balls Bx(ϵ)B^\circ_x(\epsilon) of rational radius ϵ00\epsilon \in \mathbb{Q}_{\geq 0} \subset \mathbb{R}_{\geq 0} and centered at points with rational coordinates: xnnx \in \mathbb{Q}^n \subset \mathbb{R}^n.

A compact metric space is second-countable.

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

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)[a, b).

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..

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.

If XX is second-countable and there is an open surjection f:XYf \colon X \to Y, then YY is second-countable.

For second-countable T_3 spaces X,YX, Y, if XX is locally compact, then the mapping space YXY^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 T3T_3 assumption can be removed.

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.

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

• σ\sigma-locally finite base: the topology of XX 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 AA consists precisely of all limit points of sequences in AA

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

• countably tight: for each subset AA and each point xA¯x\in \overline A there is a countable subset DAD\subseteq A such that xD¯x\in \overline D.

Implications

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

• Nagata-Smirnov metrization theorem

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

• 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 DD and a family {Uα:αA}\{U_\alpha : \alpha \in A\}, the map DαAUαAD \cap \bigcup_{\alpha \in A} U_\alpha \to A assigning dd to the unique αA\alpha \in A with dUαd \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 σ\sigma-locally finite base is first countable: obviously, every point is contained in at most countably many sets of a σ\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.