A sequential topological space is a topological space XX such that a subset AA of XX is closed if (hence iff) it contains all the limit points of all sequences of points of AA—or equivalently, such that AA is open if (hence iff) any sequence converging to a point of AA must eventually be in AA.

Equivalently, a topological space is sequential iff it is a quotient space (in TopTop) of a metric space.

Examples

Every quotient of a sequential space is sequential. In particular, every CW complex is also a sequential space. (Conversely, every sequential space is a quotient of a metrizable space, giving the alternative definition).

Properties

The category of sequential spaces is a coreflective subcategory of the category of all topological spaces.
The category of sequential spaces is a reflective subcategory of the category of subsequential spaces, much as TopTop itself is a reflective subcategory of the category of all pseudotopological spaces.
The category of sequential spaces is cartesian closed. See also convenient category of topological spaces.
The category of sequential spaces embeds fully faithfully into the category of light condensed sets (Clausen-Scholze 2023, see timestamp 9:00)

In constructive mathematics

Everything above assumes excluded middle (and probably at least countable choice). Without that, it's hard to prove the existence of any nontrivial sequential spaces.

For example, to prove that the real line is sequential as a topological space, we must find, given a set AA and a point xx such that every sequence converging to xx is eventually in AA, a positive real number δ\delta such that ]x−δ,x+δ[⊆A{]x - \delta, x + \delta[} \subseteq A, and it's not clear how to construct that number from the data at hand. (One might consider various specific sequences that converge to xx, such as (x+1/n) n(x + 1/n)_n and (x−1/n) n(x - 1/n)_n, and use them to find upper bounds on δ\delta; but no finite set of sequences will give an entire interval around xx, and proving that an infinite set of sequences that does cover an entire interval has a uniform positive upper bound on δ\delta is very tricky.)

The usual proof that the real line (or any first-countable topological space) is sequential uses excluded middle and countable choice: Supposing that AA is not open, consider x∈Ax \in A such that x∉Int(A)x \notin Int(A), pick for each δ=1/n\delta = 1/n (or for each of the countably many basic neighbourhoods U nU_n of xx in a general first-countable space) a point y ny_n such that d(x,y n)<1/nd(x,y_n) \lt 1/n (or such that y n∈U ny_n \in U_n) but y n∉Ay_n \notin A (which must exist since none of these balls/neighbourhoods are contained in AA), note that lim ny n=x\lim_n y_n = x, and get a contradiction.

For this reason, constructive analysis often requires the use of general nets (or filters) in situations where classical analysis can get by with sequences. (It is trivially true, in any topological space, that a set AA is open if every net that converges to an element xx of AA belongs eventually to AA, or equivalently that AA belongs to any filter that converges to xx; you just use the neighbourhood filter of xx.)

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 x∈A¯x\in \overline A there is a countable subset D⊆AD\subseteq A such that x∈D¯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 {U α:α∈A}\{U_\alpha : \alpha \in A\}, the map D∩⋃ α∈AU α→AD \cap \bigcup_{\alpha \in A} U_\alpha \to A assigning dd to the unique α∈A\alpha \in A with d∈U αd \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.

References

R. Engelking, General topology
Dustin Clausen, Peter Scholze, Analytic Stacks, Lecture 3 YouTube

Last revised on December 6, 2024 at 22:57:58. See the history of this page for a list of all contributions to it.