nLab
second-countable space
Second-countable spaces
- Idea
- Definitions
- Generalisations
- Examples
- Properties
- Related countability properties
- Properties
- Implications
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
[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
Generalisations
The weight of a space is the minimum of the cardinalities of the possible bases BB. 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
[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.
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.
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.