nLab sequential topological space

Idea

A topological space is sequential if (in a certain sense) you can do topology in it using only sequences instead of more general nets.

Sequential spaces are a kind of nice topological space.

Definition

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

Equivalently, a topological space is sequential iff it is a quotient space (in $Top$) 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 $Top$ 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 $A$ and a point $x$ such that every sequence converging to $x$ is eventually in $A$, a positive real number $\delta$ such that ${]x - \delta, x + \delta[} \subseteq A$, and it's not clear how to construct that number from the data at hand. (One might consider various specific sequences that converge to $x$, such as $(x + 1/n)_n$ and $(x - 1/n)_n$, and use them to find upper bounds on $\delta$; but no finite set of sequences will give an entire interval around $x$, and proving that an infinite set of sequences that does cover an entire interval has a uniform positive upper bound on $\delta$ 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 $A$ is not open, consider $x \in A$ such that $x \notin Int(A)$, pick for each $\delta = 1/n$ (or for each of the countably many basic neighbourhoods $U_n$ of $x$ in a general first-countable space) a point $y_n$ such that $d(x,y_n) \lt 1/n$ (or such that $y_n \in U_n$) but $y_n \notin A$ (which must exist since none of these balls/neighbourhoods are contained in $A$), note that $\lim_n y_n = x$, 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 $A$ is open if every net that converges to an element $x$ of $A$ belongs eventually to $A$, or equivalently that $A$ belongs to any filter that converges to $x$; you just use the neighbourhood filter of $x$.)

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

• R. Engelking, General topology

• Dustin Clausen, Peter Scholze, Analytic Stacks, Lecture 3 YouTube