Acquiring a dimension: from topology to convergence theory

Sometimes, a change of perspective reveals unexpected prospects. This was the case of the emergence of imaginary numbers within attempts of solving algebraic equations with real coefficients. A revealed universe of complex numbers had a new, metaphysical dimension, that of imaginary numbers.

Thinking of topology as of convergence of filters, rather than in terms of families of open sets or alike, was another case. Convergence theory emerged as a universe, hosting the planet of general topology. From this new perspective, topology appears laborious and austere.

In both cases, that of complex numbers and this of convergence theory, unknown objects materialized. They were often solutions of quests that were unconceivable in the old contexts. But also, surprisingly luminous solutions to tough problems manifested within the original framework. For instance, complex numbers are prodigiously instrumental in the theory of linear differential equations.

Convergence theory entirely redesigns the concepts of compactness and completeness, covers, hyperspaces, quotient and perfects maps, regularity and topologicity. It elucidates the role of sequences and that of function spaces. From this broader viewpoint, various topological explorations unveil their essential aspects and can be apprehended more consciously, and thus often more fruitfully.

In 1948 G. Choquet published a foundational paper [2], in which he noticed that the so-called Kuratowski limits cannot be topologized. From today’s perspective, they are dual convergences on hyperspaces of closed sets, which in general are not topological, because the class of topologies is not exponential (¹¹1In other terms, the category of topologies with continuous functions as morphisms is not Cartesian-closed.). The introduction of pseudotopologies by Choquet marked a turning point in our perception of general topology.

In this paper, I wish to present some salient traits of this theory, which fascinates me, and to transmit some of my enthusiasm. If you have a fancy, you may have a look at an introductory paper [5], at a textbook [8], and soon at a forthcoming book [7].

Filters arise naturally, when one considers convergence, for example, of sequences in the real line. In order to fix notation, and to introduce a new perspective, we say that a sequence $(x_{n})_{n}$ of elements of the real line $\mathbb{R}$ converges to $x$ whenever for each $\varepsilon>0$, the set $\{n:x_{n}\notin(x-\varepsilon,x+\varepsilon)\}$ is finite.

A set $V$ is a neighborhood of $x$ if there is $\varepsilon>0$ such that $(x-\varepsilon,x+\varepsilon)\subset V$. The family of all neighborhoods of $x$ is denoted by $\mathcal{N}(x)$. Let

We notice that a sequence $(x_{n})_{n}$ converges to a point $x$ if and only if

A non-empty family $\mathcal{F}$ of subsets of a given non-empty set $X$ is called a filter on $X$ if

A filter $\mathcal{F}$ is proper if $\mathrm{\varnothing}\notin\mathcal{F}$ (²²2A unique improper filter on $X$ is $2^{X}$, the family of all subsets of $X$.).

Of course, $\mathcal{N}(x)$ is a proper filter for each $x\in\mathbb{R}$, and $\mathcal{E}$ is a proper filter for each sequence $(x_{n})_{n}$. Moreover, convergence of a sequence to a point is characterized by the inclusion of filters (2.1).

Notice that in the description of convergence of sequences (³³3Here, on the real line, but it is valid in any topological space.), the order on the set of indices has vanished for a good reason that, from the convergence viewpoint, it is irrelevant. A permutation of indices of a sequence has no impact on convergence. Moreover, convergence of a sequence is invariant under arbitrary finite-to-one transformations of the set of its indices.

As we have seen in the definition of the filter $\mathcal{E}$ associated with a sequence, what counts are cofinite (that is, having finite complements) subsets of indices. And ultimately, each convergence relation is reduced to a comparison of appropriate filters.

The framework of filters appears best suited to formalize convergence. Other attempts, for instance using nets (⁴⁴4The so-called Moore-Smith convergence.) [10], have serious drawbacks.

The set of filters on a given set $X$, ordered by inclusion, is a complete lattice, in which extrema are easily described with the aid of the (bigger) lattice of all (isotone) families of subsets of $X$. Restricted to proper filters, this order is no longer an upper lattice, but it disposes of a huge set of maximal elements, called ultrafilters.

Given a non-empty set $X$, a convergence $\xi$ is any relation between (non-degenerate) filters $\mathcal{F}$ on $X$ and points of $X$ written

under the provision that $\mathcal{F}_{0}\subset\mathcal{F}_{1}$ implies $\lim\nolimits_{\xi}\mathcal{F}_{0}\subset\lim\nolimits_{\xi}\mathcal{F}_{1}$ (⁵⁵5For every filters $\mathcal{F}_{0}$ and $\mathcal{F}_{1}$ on $X$.), and $x\in\lim\nolimits_{\xi}\{x\}^{\uparrow}$ for each $x\in X$, where $\{x\}^{\uparrow}$ stands for the principal ultrafilter of $x$ (⁶⁶6$\{x\}^{\uparrow}:=\{A\subset X:x\in A\}$.).

In the convergence framework, topologies form an important subclass. Of course, a filter $\mathcal{F}$ converges to a point $x$ in a topological space, whenever each open set $O$ containing $x$ belongs to $\mathcal{F}$, in other words, whenever $\mathcal{N}(x)\subset\mathcal{F}$, as in the particular case discussed at the beginning.

Continuity is what one would expect. If $\xi$ is a convergence on $X$, and $\tau$ is a convergence on $Y$, then $f\in Y^{X}$ is continuous

whenever $x\in\lim\nolimits_{\xi}\mathcal{F}$ entails $f(x)\in\lim\nolimits_{\tau}f[\mathcal{F}]$ (⁷⁷7Where $f[\mathcal{F}]:=\{f(F):F\in\mathcal{F}\}.$ This family is not necessarily a filter on $Y$, as $f$ need not be surjective, but is a base of a filter; a subfamily $\mathcal{B}$ is said to be a base of a filter $\mathcal{G}$ if for each $G\in\mathcal{G}$ there is $B\in\mathcal{B}$ such that $B\subset G$. A convergence is obviously extended to filter-bases by $\lim\nolimits\mathcal{B}=\lim\nolimits\mathcal{G}$ if $\mathcal{B}$ is a base of $\mathcal{G}$.) for every filter $\mathcal{F}$ on $X$. Other basic constructions, known from topology, are based on the concept of continuity.

Primarily, a convergence $\zeta$ is finer than a convergence $\xi$ ($\zeta\geq\xi$) whenever the identity map $i$ is continuous $i\in C(\zeta,\xi)$ (⁸⁸8The finest convergence on $X$ is the discrete topology $\iota$, for which $x\in\lim\nolimits_{\iota}\mathcal{F}$ implies that $\mathcal{F}=\{x\}^{\uparrow}$; the coarsest one is the chaotic topology $o$, that is, $X=\lim\nolimits_{o}\mathcal{F}$ for each filter on $X$. ). The set of convergences on a given set, is a complete lattice, in which the extrema admit very simple formulae (⁹⁹9$\lim\nolimits_{\bigvee\Xi}\mathcal{F}=\bigcap_{\xi\in\Xi}\lim\nolimits_{\xi}\mathcal{F},$ and $\lim\nolimits_{\bigwedge\Xi}\mathcal{F}=\bigcup_{\xi\in\Xi}\lim\nolimits_{\xi}\mathcal{F}.$).

The initial convergence $f^{-}\tau$ is the coarsest convergence, for which $f\in C(f^{-}\tau,\tau)$. The final convergence $f\xi$ is the finest convergence on the codomain of $f$, for which $f\in C(\xi,f\xi)$. A product convergence $\prod_{j\in J}\theta_{j}$ is, of course, defined as $\bigvee_{j\in J}p_{j}^{-}\theta_{j}$, where $p_{j}$ is the projection from the product set $\prod_{i\in J}X_{i}$ onto the $j$-th component $X_{j}$ carrying the convergence $\theta_{j}$. Alike for other operations.

An elementary, though non-trivial example of non-topological convergence is

A collection $\mathbb{D}$ of filters converging to $x$ for a convergence $\xi$ is called a pavement of $\xi$ at $x$ whenever if $x\in\lim\nolimits_{\xi}\mathcal{F}$ then there is $\mathcal{D}\in\mathbb{D}$ such that $\mathcal{D}\subset\mathcal{F}$. The paving number $\mathfrak{p}(x,\xi)$ is the least cardinal such there is a pavement of $\xi$ at $x$ of cardinality $\mathfrak{p}(x,\xi)$.

In the case of topological convergences, the paving number is always equal to $1$. In the example above, it is infinite (¹²¹²12It can be shown that this convergence is not countably paved, that is, $\mathfrak{p}(x,\mathrm{Seq}\,\nu)>\aleph_{0}$ for each $x$.).

Pretopologies constitute a first generalization of topologies, and have already been considered, under various names, by Sierpiński, Čech, Hausdorff, and Choquet.

On one hand, pretopologies include topologies, which has been so far the most known and studied class of convergences. They have many similarities with topologies, and one difference: the adherence, an analogue of topological closure, is in general not idempotent. On the other hand, the class of pretopologies has a much simpler structure than the class of topologies.

A convergence is a pretopology if for each point (of the underlying set), there is a coarsest filter converging to that point. Therefore, a convergence is a pretopology if and only if its paving number is $1$.

For an arbitrary convergence $\theta$, the filter

is called the vicinity filter of $\theta$ at $x$.

Consequently, each topology $\xi$ is a pretopology, and if it is, then $\mathcal{V}_{\xi}(x)=\mathcal{N}_{\xi}(x)$ is the neighborhood filter of $\xi$ at $x$.

If $\mathcal{A}$ is a family of subsets of $X$, then the grill of $\mathcal{A}$ is defined by

For an arbitrary convergence $\theta$, the adherence $\operatorname{adh}\nolimits_{\theta}A$ of a set $A$ can be defined by

For any convergence $\theta$, a set $A$ is called $\theta$-closed if $\operatorname{adh}\nolimits_{\theta}A\subset A$. The $\theta$-closure is defined by $\operatorname{cl}\nolimits_{\theta}A:=\bigcap_{H\supset A}\{H:\operatorname{adh}\nolimits_{\theta}H\subset H\}$.

Notice that if $\xi$ is a pretopology, then by (set-adherence) its adherence determines all its vicinity filter, hence its convergent filters.

In other terms, a pretopology $\xi$ is topological whenever $\operatorname{adh}\nolimits_{\xi}(\operatorname{adh}\nolimits_{\xi}A)\subset\operatorname{adh}\nolimits_{\xi}A$ for each $A$. In this case, $\operatorname{adh}\nolimits_{\xi}A$ is, of course, $\xi$-closed and and equal to $\operatorname{cl}\nolimits_{\xi}A$, the closure of $A$.

Topologies, pretopologies, and many other fundamental classes of convergences are projective. This means that for each convergence $\theta$, there exists a finest topology $J\theta$ among the topologies that are coarser than $\theta$. It is easy to see that so defined $J$ is concrete ($\left|J\theta\right|=\left|\theta\right|$), increasing ($\theta_{0}\leq\theta_{1}$ implies $J\theta_{0}\leq J\theta_{1}$), idempotent ($J(J\theta)=J\theta$), as well as descending ($J\theta\leq\theta$) (¹³¹³13for each convergences $\theta,\theta_{0},\theta_{1}$). If $J$ preserves continuity, that is, if $C(\xi,\tau)\subset C(J\xi,J\tau)$ for any convergences $\xi$ and $\tau$, then $J$ is a concrete functor (¹⁴¹⁴14Basic facts from category theory are used here instrumentally, so to say, objectwise. Functors are certain maps defined on classes of morphisms, and then specialized to the classes of objects viewed as identity morphisms. Because the category of convergences with continuous maps as morphisms is concrete over the category of sets, it is enough to define concrete functors merely on objects. ).

Topologies an pretopologies are reflective; the reflector $\mathrm{T}$ on the class of topologies is called the topologizer, the reflector $\mathrm{S}_{0}$ on the class of pretopologies is called the pretopologizer. Both admit similar explicit descriptions

The objectwise use of functors associated with various classes of convergences, constitutes a sort of calculus, enabling to perceive in a unified way miscellaneous aspects of convergences, hence in particular of topologies.

Let $\xi$ be a convergence on $X$, and let $\mathcal{A}$ be a family of subsets of $X$. The adherence $\operatorname{adh}\nolimits_{\xi}\mathcal{A}$ is defined by

In particular, if $A\subset X$, then $\operatorname{adh}\nolimits_{\xi}A=\operatorname{adh}\nolimits_{\xi}\{A\}$, which is the set-adherence, already introduced in (set-adherence). It is straightforward that, for each filter $\mathcal{F}$ on $X$,

We denote by $\mathbb{F}$ the class of all filters, by $\mathbb{F}_{1}$ the class of countably based filters, and by $\mathbb{F}_{0}$ the class of principal (or finitely based) filters. By convention, if $\mathbb{F}_{0}\subset\mathbb{H}\subset\mathbb{F}$ then $\mathbb{H}X$ is the set of filters on $X$ that belong to the class $\mathbb{H}$ (¹⁵¹⁵15In particular, the filters from $\mathbb{F}_{0}X$ are of the form $A^{\uparrow}:=\{F\subset X:A\subset F\}$ for some $A\subset X$.).

Let $\mathbb{H}$ be a class of filters. We say that $\mathbb{H}$ is initial if $f^{-}[\mathcal{H}]\in\mathbb{H}$ for each $\mathcal{H}\in\mathbb{H}$, final if $f[\mathcal{H}]\in\mathbb{H}$ for each $\mathcal{H}\in\mathbb{H}$ (¹⁶¹⁶16Of course, it is understood that if $f\in Y^{X}$ and $\mathcal{H}\in\mathbb{H}Y$ then $f^{-}[\mathcal{H}]\in\mathbb{H}X$, and if $\mathcal{H}\in\mathbb{H}X$ then $f[\mathcal{H}]\in\mathbb{H}Y$.).

Assume that $\mathbb{H}$ is an initial class of filters. Then a convergence $\xi$ is called $\mathbb{H}$-adherence-determined if $\lim\nolimits_{\xi}\mathcal{F}\supset\bigcap\nolimits_{\mathbb{H}\ni\mathcal{H}\subset\mathcal{F}^{\#}}\operatorname{adh}\nolimits_{\xi}\mathcal{H}.$ The class of $\mathbb{H}$-adherence-determined convergences is concretely reflective, and the reflector $A_{\mathbb{H}}$ fulfills

that is, $x\in\lim\nolimits_{A_{\mathbb{H}}\xi}\mathcal{F}$ provided that $x\in\operatorname{adh}\nolimits_{\xi}\mathcal{H}$ for each $\mathcal{H}\in\mathbb{H}$ such that $\mathcal{H}\subset\mathcal{F}^{\#}$.

Since we assume that $\mathbb{F}_{0}\subset\mathbb{H}\subset\mathbb{F}_{1}$, the $\mathbb{F}_{0}$-adherence-determined pretopologies form the narrowest class, and $\mathbb{F}$-adherence-determined pseudotopologies the largest. By the way, topologies are not adherence-determined.

We shall yet consider two intermediate classes: paratopologies, corresponding to for which $\mathrm{S}_{1}=A_{\mathbb{F}_{1}}$, and hypotopologies, corresponding to the class $\mathbb{F}_{\wedge 1}$ of countably deep filters (¹⁷¹⁷17That is, the filters $\mathcal{F}$ such that $\mathcal{F}_{0}\subset\mathcal{F}$ and $\mathrm{card\,}\mathcal{F}_{0}$ is countable, then $\bigcap\mathcal{F}_{0}\in\mathcal{F}.$ ), for which $\mathrm{S}_{\wedge 1}=A_{\mathbb{F}_{\wedge 1}}$. It was recently observed [16] that pretopologies constitute the intersection of paratopologies an hypotopologies.

By the way, as all functors, the adherence-determined reflectors preserve continuity, but moreover fulfill

for each $f$ and $\tau$. In particular, if $f$ is an injection, then (5.1) means that $A_{\mathbb{H}}$ commutes with the construction of subspaces for $\mathbb{F}_{0}\subset\mathbb{H}\subset\mathbb{F}_{1}$! Mind that the topologizer $\mathrm{T}$ is not of the form $A_{\mathbb{H}}$, and does not commute with the construction of subspaces.

We have seen that pseudotopologies constitute an adherence-determined class with respect to the class $\mathbb{F}$ of all filters. It easily follows from the definition that if $\xi$ is a pseudotopology, then

where $\beta\mathcal{F}$ stands for the set of ultrafilters that are finer than a filter $\mathcal{F}$. It is remarkable that the pseudotopologizer $\mathrm{S}$ commutes with arbitrary products, that is, if $\Xi$ is a set of convergences, then

This is because $\mathrm{S}$ commutes with construction of initial convergence (as an adherence-determined reflector), and also with arbitrary suprema (¹⁸¹⁸18Let us first prove that $\mathrm{S}(\bigvee\Xi)=\bigvee\nolimits_{\xi\in\Xi}\mathrm{S}\xi.$ Indeed, $\lim\nolimits_{\mathrm{S}(\bigvee\Xi)}\mathcal{F}=\bigcap\nolimits_{\mathcal{U}\in\beta\mathcal{F}}\lim\nolimits_{\bigvee\Xi}\mathcal{U}$, which is equal to $=\bigcap\nolimits_{\mathcal{U}\in\beta\mathcal{F}}\bigcap_{\xi\in\Xi}\lim\nolimits_{\xi}\mathcal{U}$. By commuting the intersections, we get $\bigcap_{\xi\in\Xi}\bigcap\nolimits_{\mathcal{U}\in\beta\mathcal{F}}\lim\nolimits_{\xi}\mathcal{U=}\bigcap_{\xi\in\Xi}\lim\nolimits_{\mathrm{S}\xi}\mathcal{U}=\lim\nolimits_{\bigvee_{\xi\in\Xi}\mathrm{S}\xi}\mathcal{F}$. By the definition of product, $\prod\Xi=\bigvee\nolimits_{\xi\in\Xi}p_{\xi}^{-}\xi$, where $p_{\xi}:\prod_{\zeta\in\Xi}\left|\zeta\right|\longrightarrow\left|\xi\right|$ is the $\xi$-projection. Therefore, by (5.1), $\mathrm{S}(\prod\Xi)=\mathrm{S}(\bigvee\nolimits_{\xi\in\Xi}p_{\xi}^{-}\xi)=\bigvee\nolimits_{\xi\in\Xi}\mathrm{S}(p_{\xi}^{-}\xi)=\bigvee\nolimits_{\xi\in\Xi}p_{\xi}^{-}(\mathrm{S}\xi)=\prod\nolimits_{\xi\in\Xi}\mathrm{S}\xi$.).

Although $F(\xi_{0}\times\xi_{1})\geq F\xi_{0}\times F\xi_{1}$ holds for each functor $F$, the converse is rather an exception. For instance, the pretopologizer $\mathrm{S}_{0}$ commutes with the construction of initial convergence (like the pseudotopologizer $\mathrm{S}$), but not with suprema, hence not with products.

A first example is that of quotient maps. In topology, a continuous map $f\in C(\xi,\tau)$ (between topologies $\xi$ and $\tau$) is said to be quotient if $\tau\geq\mathrm{T}(f\xi)$, hence, because of the continuity assumption, $\tau=\mathrm{T}(f\xi)$. Let us remark that, if $\xi$ is a topology, that is, $\mathrm{T}\xi=\xi$, then the final convergence $f\xi$ need not be a topology; actually, it can be, so to say, almost anything, as each finitely deep convergence (¹⁹¹⁹19A convergence $\theta$ is called finitely deep if $\lim\nolimits_{\theta}\mathcal{F}_{0}\cap\lim\nolimits_{\theta}\mathcal{F}_{1}\subset\lim\nolimits_{\theta}(\mathcal{F}_{0}\cap\mathcal{F}_{1})$ for any $\mathcal{F}_{0}$ and $\mathcal{F}_{1}$.) is a convergence quotient of topologies.

It has long been known that quotient maps preserve some properties, like sequentiality, but do not preserve others, like Fréchetness. For this reason, numerous quotient-like maps (quotient, hereditarily quotient, countably biquotient, biquotient, triquotient, almost open) and their preservation properties were intensively investigated. In his [11], E. Michael gathered, generalized, and refined numerous preservation existent results (A. V. Arhangel’skii [1], V. I. Ponomarev [17], S. Hanai [9], F. Siwiec [18], and others) for these quotient-like maps. Richness and complexity of these investigations made of this quotient quest a veritable jungle (²⁰²⁰20A metaphor came, when I tackled to decorticate this article. I realized that I would not grasp its underlying ideas, unless I transform the jungle into an Italian garden. I evoked it during a conference in honor of Peter Collins and Mike Reed in Oxford in 2006, and Ernest Michael, who attended, appreciated.).

Using convergence-theoretic methods [3], it was possible to figure out that virtually all these quotient-like maps follow the same pattern, namely they are of the form

where $J$ is a reflector on a subclass of convergences, for a map $f\in C(\xi,\tau)$ (²¹²¹21In fact, in various problems the continuity of a quotient-like map is inessential, and can be dropped.), a panorama that was unavailable within the framework of topologies. In particular, a map $f$ fulfilling ($J$-quotient map) is quotient if $J=\mathrm{T}$, hereditarily quotient if $J=\mathrm{S_{0}}$ (pretopologizer), countably biquotient if $J=\mathrm{S_{1}}$ (paratopologizer), biquotient if $J=\mathrm{S}$ (pseudotopologizer), and almost open if $J=\mathrm{I}$ (identity functor). By the way, it is often handy to say $\mathbb{H}$-quotient instead of $A_{\mathbb{H}}$-quotient.

Biquotient maps are the only among the listed classes that are preserved by arbitrary products, which, of course, is due to (commutation).

Of course, ($J$-quotient map) transcends topologies, but when limited to topologies $\xi$ and $\tau$, it yields a topological conclusion, having passed beyond. The name hereditarily quotient, traditonally used in the topological context, is due to the fact that each restriction of a $\mathrm{S}_{0}$-quotient map remains a $\mathrm{S}_{0}$-quotient (²²²²22Indeed, let $\xi$ be a convergence on $X$, and $\tau$ be a convergence on $Y$. If $f\in Y^{X}$ fulfills $\tau\geq\mathrm{S}_{0}(f\xi)$, then for $B\subset Y$ and the injection $j_{B}\in Y^{B}$, $$j_{B}^{-}\tau\geq j_{B}^{-}\mathrm{S}_{0}(f\xi)=\mathrm{S}_{0}(j_{B}^{-}\circ f)\xi,$$ and the final convergence of $\xi$ by $j_{B}^{-}\circ f$ is equal to the final convergence of $j_{f^{-}(B)}^{-}\xi$ by $j_{B}^{-}\circ f$.).

It turns out that sundry properties, like sequentiality, Fréchetness, local compactness, and so on, appear as solutions $\theta$ of functorial inequalities of the type

where $J$ is a concrete reflector, and $E$ is an appropriate concrete coreflector, that is a concrete, increasing, idempotent and ascending ($\theta\leq E\theta$) functor (compare with the definition of concrete reflector).

Now a preservation scheme becomes manifest (²³²³23Let $\xi\geq JE\xi$, ($J$-quotient map) and $f\in C(\xi,\tau)$. Then $f\xi\geq f(JE\xi)\geq JE(f\xi)$, the last inequality being consequence of $f(F\xi)\geq F(f\xi$), valid for each functor $F$. By ($J$-quotient map) and idempotency of $J$, we infer $\tau\geq J(f\xi)\geq JE(f\xi)\geq JE\tau$, the last inequality entailed by continuity: $f\xi\geq\tau$. As a result, $\tau\geq JE\tau$.).

Let us illustrate the preservation result above by the two properties discussed in the example. A more exhaustive list of special cases of this theorem, can be found in [3], where all of 20 entries correspond to theorems, many of which were demonstrated in numerous papers. See also [8, p. 400].

Here is a simple proof of sufficiency (²⁴²⁴24Let $\sigma=J\sigma$. By definition, (duality) $$\xi\times[\xi,\sigma]\geq\mathrm{ev}^{-}\sigma,$$ and $[\xi,\sigma]$ is the coarsest convergence, for which the inequality above holds. If $J$ commutes with finite products then, from (duality), $$\xi\times J[\xi,\sigma]\geq J\xi\times J[\xi,\sigma]\geq J(\xi\times[\xi,\sigma])\geq J(\mathrm{ev}^{-}\sigma)\geq\mathrm{ev}^{-}(J\sigma)=\mathrm{ev}^{-}\sigma,$$ the last inequality following from $F(f\sigma)\geq f^{-}(F\sigma)$, valid for each functor $F$. As, by assumption, $[\xi,\sigma]$ is the coarsest convergence fulfilling (duality), $J[\xi,\sigma]\geq[\xi,\sigma]$, hence $J[\xi,\sigma]=[\xi,\sigma]$, because $J$ is a reflector.).

We understand now why the Kuratowski convergence on the hyperspaces of closed sets, considered by Choquet in [2], is not topological (²⁵²⁵25Indeed, if $\tau$ is a topology on a set $X$, then the (upper) Kuratowski convergence on the hyperspace consisting of all $\tau$-closed sets is $[\tau,\$]$, where the Sierpiński topology $\$$ on $\{0,1\}$, the closed sets of which are $\mathrm{\varnothing},\{0\},$ and $\{0,1\}$. The hyperspace can be identified with $C(\tau,\$)$, the space of continuous from $\tau$ to $\$$. Accordingly, $A$ is $\tau$-closed if and only if the characteristic function $\chi_{A}\in\{0,1\}^{X}$, that is, $A=\{x\in X:\chi_{A}(x)=1\}$, fulfills $\chi_{A}\in C(\tau,\$)$.).

Given any reflector $J$, there exists the least exponential reflector $\mathrm{Epi}^{J}$ such that $J\leq\mathrm{Epi}^{J}$ (²⁶²⁶26It follows that $J$ is exponential if and only if $J=\mathrm{Epi}^{J}$.). The corresponding least exponential reflective class $\mathrm{fix}(\mathrm{Epi}^{J})$ including $\mathrm{fix}(J)$, is called the exponential hull of $\mathbf{J}$. Duality theory, developed by F. Mynard [12, 13], and others, allows to characterize exponential hulls.

A construction uses the $\sigma$-dual convergence $[[\xi,\sigma],\sigma]$ of the $\sigma$-dual convergence $[\xi,\sigma]$ of $\xi$, which is a convergence on $C([\xi,\sigma],\sigma)$. Then $\mathrm{Epi}^{\sigma}\xi=j^{-}[[\xi,\sigma],\sigma]$, that is, the initial convergence of the $\sigma$-bidual convergence by the natural injection $j:X\longrightarrow Z^{(Z^{X})}$, which turns out to be continuous: $j\in C(\xi,[[\xi,\sigma],\sigma])$. Finally,

It turns out that the two most important non-topological convergences, introduced by G. Choquet, are intimately related by duality.

See also (²⁷²⁷27Let us mention that $\mathrm{Epi}^{\mathrm{S}_{0}}\xi=\mathrm{S\xi}=j^{-}[[\xi,\yen],\yen]$, where $\yen$ is the Bourdaud pretopology. The Bourdaud pretopology $\yen$ is defined on $\{0,1,2\}$ by convergence of ultrafilters as follows $$\lim\nolimits_{\yen}\{0\}^{\uparrow}=\{0,1\},\;\lim\nolimits_{\text{\textyen}}\{1\}^{\uparrow}=\{0,1,2\},\;\lim\nolimits_{\text{\textyen}}\{2\}^{\uparrow}=\{0,1,2\}.$$ The exponential hull of topologies is the class of epitopologies, defined by P. Antoine, and then $\mathrm{Epi}^{\mathrm{T}}\xi=j^{-}[[\xi,\$],\$]$.).

A subset $A$ of a topological space is called compact if every open cover of $A$ admits a finite subcover of $A$, equivalently, each ultrafilter on $A$ has a limit point in $A$, or else, each filter on $A$ has an adherence point in $A$. Many authors require that, besides, the topology be Hausdorff.

A convergence is, in general, not determined by its open sets, and thus open covers are not an adequate concept in this context. A natural extension to convergence spaces of the notion of cover is used to define cover-compact sets. The point is that cover-compactness and filter-compactness are no longer equivalent for general convergences. Moreover, it turns out that cover-compactness is not preserved under continuous maps.

Specializing the definition above to a topology $\xi$ on a set $X$, we infer that $\mathcal{P}$ is a $\xi$-cover of $A$ if and only if $A\subset\bigcup\nolimits_{P\in\mathcal{P}}\operatorname{inh}\nolimits_{\xi}P$, where $\operatorname{inh}\nolimits_{\xi}P:=X\setminus\operatorname{adh}\nolimits_{\xi}(X\setminus A)$ is the $\xi$-inherence of $P$.

Endowed with this extended concept of cover, we are in a position to discuss cover-compactness for general convergences.

The following simple (²⁸²⁸28A family $\mathcal{P}$ is not a $\xi$-cover of a set $A$, whenever there exists a filter $\mathcal{F}$ such that $A\cap\lim\nolimits_{\xi}\mathcal{F}\neq\varnothing$ and $P\notin\mathcal{F}$ for each $P\in\mathcal{P}$. In other words, $F\cap P^{c}=F\setminus P\neq\varnothing$ for each $P\in\mathcal{P}$ and each $F\in\mathcal{F}$, equivalently, $\mathcal{P}_{c}\#\mathcal{F}$, that is, $A\cap\operatorname{adh}\nolimits_{\xi}\mathcal{P}_{c}\neq\varnothing$.), but very consequential observation [4] enables to easily compare the two variants.

To this end, we focus on ideal covers. A family of subsets of a given set is called an ideal if

Clearly, $\mathcal{P}$ is an ideal of subsets of $X$ if and only if $\mathcal{P}_{c}$ is a filter on $X$. Passing from arbitrary covers to ideal covers makes no difference in topology, but does make in general. By the preceding proposition, on setting $\mathcal{H}=\mathcal{P}_{c}$, we characterize filter-compactness in terms of ideal covers:

Cover-compactness implies (filter)-compactness for pretopologies. Indeed, if $\xi$ is a pretopology, and $A$ is $\xi$-cover-compact, then in particular, for each ideal $\xi$-cover $\mathcal{P}$ of $A$, there exists a finite $\mathcal{P}_{0}\subset\mathcal{P}$ such that $A\subset\bigcup\nolimits_{P\in\mathcal{P}_{0}}\operatorname{inh}\nolimits_{\xi}P\subset\operatorname{inh}\nolimits_{\xi}\bigcup\nolimits_{P\in\mathcal{P}_{0}}P\subset\bigcup\nolimits_{P\in\mathcal{P}_{0}}P\in\mathcal{P}$, because $\mathcal{P}$ is an ideal. Hence $A\in\mathcal{P}$, so that $A$ is $\xi$-compact.

On the other hand, there exist pretopologies, where the two notions differ [8, Example IX.11.8].

Moreover, each finite set is $\xi$-compact for any convergence $\xi$ (²⁹²⁹29In fact, if a filter $\mathcal{H}$ fulfills $\{x\}\in\mathcal{H}^{\#}$ then $x\in\bigcap\mathcal{H}$, and thus $x\in\operatorname{adh}\nolimits_{\xi}\mathcal{H},$ equivalently $\{x\}\cap\operatorname{adh}\nolimits_{\xi}\mathcal{H}\neq\mathrm{\varnothing}$.), but

Compact families of sets generalize both compact sets and convergent filters, and this generalization is not just a whim. It has important applications, and, perhaps more importantly, evidences mathematical laws that remained invisible on the level of compactness of sets.

Let $\xi$ be a convergence on $X$. A family $\mathcal{A}$ of subsets of $X$ is said to be $\xi$-compact at a family $\mathcal{B}$ of subsets of $X$ if, for each filter $\mathcal{H}$,

In particular, $\mathcal{A}$ is called $\xi$-compact if it is $\xi$-compact at itself; $\xi$-compactoid if it is $\xi$-compact at $X$ (³¹³¹31The set $\kappa(\xi)$ of all $\xi$-compact (isotone) families on $X=\left|\xi\right|$ fulfills: $\mathrm{\varnothing},2^{X}\in\kappa(\xi)$, $\{\mathcal{A}_{j}:j\in J\}\subset\kappa(\xi)$ entails $\bigcup_{j\in J}\mathcal{A}_{j}\in\kappa(\xi)$, and $\bigcap_{j\in J}\mathcal{A}_{j}\in\kappa(\xi)$, whenever $J$ is finite. In other words, $\kappa(\xi)$ has the properties of a family of open sets of a topology on $2^{X}$.).

It is clear that a subset $A$ of $X$ is $\xi$-compact ($\xi$-compactoid), whenever $A^{\uparrow}:=\{F\subset X:A\subset F\}$ is (¹⁵¹⁵footnotemark: 15). On the other hand, it is straightforward that $\mathcal{F}$ is $\xi$-compact at $\{x\}$ if and only if $x\in\lim\nolimits_{\mathrm{S}\xi}\mathcal{F}$. Incidentally, it is straightforward that $\xi$-compactness and $\mathrm{S}\xi$-compactness coincide.

This simple fact prefigures the pseudotopological nature of compactness, which will be evidenced in a moment.

At this point, it will be instrumental to consider again the notion of grill, from a somewhat different perspective. Recall that $\mathcal{A}^{\#}:=\bigcap_{A\in\mathcal{A}}\{H\subset X:A\cap H\neq\mathrm{\varnothing}\}$ for a family $\mathcal{A}$ of subsets of $X$. Now, for another family $\mathcal{H}$ on $X$, the condition $\mathcal{H}\subset\mathcal{A}^{\#}$ is equivalent to $\mathcal{A}\subset\mathcal{H}^{\#}$, so we denote this relation symmetrically, by $\mathcal{H}\#\mathcal{A}$ (³²³²32Of course, $\mathcal{H}\#\mathcal{A}$ whenever $H\cap A\neq\mathrm{\varnothing}$ for each $H\in\mathcal{H}$ and $A\in\mathcal{A}$.). If $\mathcal{A}$ is on $X$, and $\mathcal{B}$ is on $Y$, and $f:X\longrightarrow Y$, then it is easy to see that

For a given convergence $\xi$ on $X$, define the associated characteristic convergence $\chi_{\xi}$ by

It is immediate that, for a set $\Xi$ of convergences,

As an immediate consequence of this lemma and of (commutation),

If we restrict filters $\mathcal{H}$ in (compact family) to a class $\mathbb{H}$ of filters, then we obtain a notion of $\mathbb{H}$-compactness. We assume that $\mathbb{F}_{0}\subset\mathbb{H}\subset\mathbb{F}$, that is, that the said class includes all principal filters. $\mathcal{A}$ is said to be $\xi$-$\mathbb{H}$-compact at $\mathcal{B}$ if

Some instances of this notion have been already known in topological context, like $\mathbb{F}_{1}$-compactness, that is, countable compactness (³³³³33By the way, sequential compactness of $\xi$ coincides with $\mathbb{F}_{1}$-compactness of $\mathrm{Seq}\xi$.), or $\mathbb{F}_{\wedge 1}$-compactness, that is, Lindelöf property. If $H=A_{\mathbb{H}}$ is the $\mathbb{H}$-adherence-determined reflector, then a filter $\mathcal{F}$ is $\mathbb{H}$-compactoid for $\xi$, whenever (³⁴³⁴34Recall that $x\in\lim\nolimits_{H\xi}\mathcal{F}$ whenever $\mathcal{F}$ is $\xi$-$H$-compact at $\{x\}$.)

Of course, once established for special reflectors, the formula above can be used as definition of $H$-compactness for arbitrary reflectors $H$.

Observe that, for other refectors $H$ than the pseudotopologizer, $H$-compactness is nor preserved even by finite products if $H$ does not commute with such products.

A step further is to extend $\mathbb{H}$-compactness to relations. Roughly speaking (³⁵³⁵35Let $\theta$ be a convergence on $W$ and $\sigma$ be a convergence on $Z$. A relation $R$ is called $\mathbb{J}$-compact if $w\in\lim\nolimits_{\theta}\mathcal{F}$ implies that $R(w)\#\operatorname{adh}\nolimits_{\sigma}\mathcal{H}$ for each $\mathcal{H}\#R[\mathcal{F}]$ such that $\mathcal{H}\in\mathbb{J}$.), a relation $R$ is $\mathbb{H}$-compact if $y\in\lim\nolimits\mathcal{F}$ implies that $R[\mathcal{F}]$ is $\mathbb{H}$-compact at $Ry$. Continuous maps and various quotient maps can be characterized in terms of compact relations [14]. However, most advantageous applications of compact relations are to various perfect-like maps.

A surjective map $f$ is $\mathbb{H}$-perfect if and only if the relation $f^{-}$ is $\mathbb{H}$-compact.

For instance, $\mathbb{F}$-perfect maps are precisely perfect maps are close maps with compact fibers. $\mathbb{F}_{1}$-perfect maps, or countably perfect maps are close maps with countably compact fibers.

In particular, arbitrary product of $\mathbb{F}$-perfect maps is $\mathbb{F}$-perfect. This is because the pseudotopologizer $\mathrm{S}=A_{\mathbb{F}}$ commutes with arbitrary products, or else because the product of compact fiber relations is compact by the Generalized Tikhonov Theorem.

Perfect-like and quotient-like properties embody various degrees of converse stability of maps $f$, or in other terms, of stability of fiber relations $f^{-}$, which is the inverse relation of $f$. Let us rewrite these properties in expanded form, where $f:\left|\xi\right|\longrightarrow\left|\tau\right|$.

A surjective map $f$ is $\mathbb{H}$-quotient if and only if

holds for each $\mathcal{H}\in\mathbb{H}$. A surjective map $f$ is $\mathbb{G}$-perfect if and only if

holds for each $\mathcal{G}\in\mathbb{G}$.

No arrow can be reversed. Indeed,

I hope that these outlines allow to grasp the essence of convergence theory. Sure enough, only few aspects have been touched upon, and most remain beyond this presentation.

For example, various types of compactness are instances of numerous kinds of completeness. The completeness number $\mathrm{compl}(\xi)$ of a convergence $\xi$ is the least cardinality of a collection of $\xi$-non-adherent filters that fill the set of $\xi$-non-convergent ultrafilters in the Stone space. This way, compact convergences $\xi$ are characterized by $\mathrm{compl}(\xi)=0,$ locally compactoid by $\mathrm{compl}(\xi)<\infty$, and topologically complete by $\mathrm{compl}(\xi)\leq\aleph_{0}$. Each convergence has its completeness number; for the “very incomplete” space of rational numbers, this number is the dominating number $\mathfrak{d}$.

It was shown in this paper that a generalization of Tikhonov Theorem is a simple corollary of the commutation of the pseudotopologizer with arbitrary products. It turns out that it is also a simple consequence of a theorem on the completeness number of products [6][8].

It appears that $\mathrm{compl}(\xi)$ is equal to the (free) pseudo-paving number of the dual convergence $[\xi,\$]$ at $\mathrm{\varnothing}$, and the (free) paving number of $[\xi,\$]$ at $\mathrm{\varnothing}$ is equal to the ultra-completeness number of $\xi$ [15].

We see that, in the framework of topologies, it would be impossible to consider a property dual to Čech completeness (countable completeness number), because the paving and pseudo-paving numbers of a topology do not exceed $1$.

I could long display similar examples, but I expect that these few exhibited in this paper would convince you of the interest of convergence theory.