Acquiring a dimension: from topology to convergence theory
AACQUIRING A DIMENSION:FROM TOPOLOGY TO CONVERGENCE THEORY
SZYMON DOLECKI
Abstract.
Convergence theory is an extension of general topology. In con-trast with topology, it is closed under some important operations, like exponen-tiation. With all its advantages, convergence theory remains rather unknown.It is an aim of this paper to make it more familiar to the mathematical com-munity. Introduction
Sometimes, a change of perspective reveals unexpected prospects. This was thecase of the emergence of imaginary numbers within attempts of solving algebraicequations with real coefficients. A revealed universe of complex numbers had anew, metaphysical dimension, that of imaginary numbers.Thinking of topology as of convergence of filters, rather than in terms of familiesof open sets or alike, was another case. Convergence theory emerged as a universe,hosting the planet of general topology. From this new perspective, topology appearslaborious and austere.In both cases, that of complex numbers and this of convergence theory, unknownobjects materialized. They were often solutions of quests that were unconceivablein the old contexts. But also, surprisingly luminous solutions to tough problemsmanifested within the original framework. For instance, complex numbers are prodi-giously instrumental in the theory of linear differential equations.Convergence theory entirely redesigns the concepts of compactness and complete-ness, covers, hyperspaces, quotient and perfects maps, regularity and topologicity.It elucidates the role of sequences and that of function spaces. From this broaderviewpoint, various topological explorations unveil their essential aspects and canbe apprehended more consciously, and thus often more fruitfully.In 1948
G. Choquet published a foundational paper [2], in which he noticedthat the so-called
Kuratowski limits cannot be topologized. From today’s perspec-tive, they are dual convergences on hyperspaces of closed sets, which in general arenot topological, because the class of topologies is not exponential ( ). The intro-duction of pseudotopologies by Choquet marked a turning point in our perceptionof general topology.In this paper, I wish to present some salient traits of this theory, which fascinatesme, and to transmit some of my enthusiasm. If you have a fancy, you may have
Date : June 23, 2020.2000
Mathematics Subject Classification.
Key words and phrases.
Filter convergence, pseudotopology, pretopology, topology, quotientmap, perfect map, dual convergence, exponential reflective hull. In other terms, the category of topologies with continuous functions as morphisms is notCartesian-closed. a r X i v : . [ m a t h . GN ] J un SZYMON DOLECKI a look at an introductory paper [5], at a textbook [8], and soon at a forthcomingbook [7]. 2.
The choice of filters
Filters arise naturally, when one considers convergence, for example, of sequencesin the real line. In order to fix notation, and to introduce a new perspective, wesay that a sequence ( x n ) n of elements of the real line R converges to x wheneverfor each ε > , the set { n : x n / ∈ ( x − ε, x + ε ) } is finite.A set V is a neighborhood of x if there is ε > such that ( x − ε, x + ε ) ⊂ V . Thefamily of all neighborhoods of x is denoted by N ( x ) . Let(sequential filter) E := { E ⊂ R : { n : x n / ∈ E } is finite } . We notice that a sequence ( x n ) n converges to a point x if and only if(2.1) N ( x ) ⊂ E . A non-empty family F of subsets of a given non-empty set X is called a filter on X if ( F ∈ F ) ∧ ( F ∈ F ) ⇐⇒ F ∩ F ∈ F . A filter F is proper if ∅ / ∈ F ( ).Of course, N ( x ) is a proper filter for each x ∈ R , and E is a proper filter for eachsequence ( x n ) n . Moreover, convergence of a sequence to a point is characterized bythe inclusion of filters (2.1).Notice that in the description of convergence of sequences ( ), the order on theset 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 conver-gence. Moreover, convergence of a sequence is invariant under arbitrary finite-to-onetransformations of the set of its indices.As we have seen in the definition of the filter E associated with a sequence, whatcounts are cofinite (that is, having finite complements) subsets of indices. Andultimately, each convergence relation is reduced to a comparison of appropriatefilters.The framework of filters appears best suited to formalize convergence. Otherattempts, for instance using net s ( ) [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 nolonger an upper lattice, but it disposes of a huge set of maximal elements, called ultrafilters . 3. Convergences
Given a non-empty set X , a convergence ξ is any relation between (non-degenerate)filters F on X and points of X written x ∈ lim ξ F , A unique improper filter on X is X , the family of all subsets of X . Here, on the real line, but it is valid in any topological space. The so-called
Moore-Smith convergence . CQUIRING A DIMENSION: FROM TOPOLOGY TO CONVERGENCE THEORY 3 under the provision that F ⊂ F implies lim ξ F ⊂ lim ξ F ( ), and x ∈ lim ξ { x } ↑ for each x ∈ X , where { x } ↑ stands for the principal ultrafilter of x ( ).In the convergence framework, topologies form an important subclass. Of course,a filter F converges to a point x in a topological space, whenever each open set O containing x belongs to F , in other words, whenever N ( x ) ⊂ F , as in the particularcase discussed at the beginning. Continuity is what one would expect. If ξ is a convergence on X , and τ is aconvergence on Y , then f ∈ Y X is continuous f ∈ C ( ξ, τ ) whenever x ∈ lim ξ F entails f ( x ) ∈ lim τ f [ F ] ( ) for every filter F on X . Otherbasic constructions, known from topology, are based on the concept of continuity.Primarily, a convergence ζ is finer than a convergence ξ ( ζ ≥ ξ ) whenever theidentity map i is continuous i ∈ C ( ζ, ξ ) ( ). The set of convergences on a given set,is a complete lattice, in which the extrema admit very simple formulae ( ).The initial convergence f − τ is the coarsest convergence, for which f ∈ C ( f − τ, τ ) .The final convergence f ξ is the finest convergence on the codomain of f , for which f ∈ C ( ξ, f ξ ) . A product convergence (cid:81) j ∈ J θ j is, of course, defined as (cid:87) j ∈ J p − j θ j ,where p j is the projection from the product set (cid:81) i ∈ J X i onto the j -th component X j carrying the convergence θ j . Alike for other operations.An elementary, though non-trivial example of non-topological convergence is Example.
The sequential modification
Seq ν of a usual topology of the real line ν , in which x ∈ lim Seq ν F provided that there exists a sequence ( x n ) n such that lim n →∞ x n = x , and { x k : k > n } ∈ F for each n ( ). In other terms, a filterconverges to a point whenever it is finer than a sequential filter converging to thatpoint. However, there exists no coarsest filter converging to a given point. Infact, the infimum of all sequential filters converging to x is the neighborhood filter N ν ( x ) of x for the usual topology, but x / ∈ lim Seq ν N ν ( x ) , because each V ∈ N ν ( x ) is uncountable ( ).A collection D of filters converging to x for a convergence ξ is called a pavement of ξ at x whenever if x ∈ lim ξ F then there is D ∈ D such that D ⊂ F . The pavingnumber p ( x, ξ ) is the least cardinal such there is a pavement of ξ at x of cardinality p ( x, ξ ) .In the case of topological convergences, the paving number is always equal to .In the example above, it is infinite ( ). For every filters F and F on X . { x } ↑ := { A ⊂ X : x ∈ A } . Where f [ F ] := { f ( F ) : F ∈ F} . This family is not necessarily a filter on Y , as f need notbe surjective, but is a base of a filter; a subfamily B is said to be a base of a filter G if for each G ∈ G there is B ∈ B such that B ⊂ G . A convergence is obviously extended to filter-bases by lim B = lim G if B is a base of G . The finest convergence on X is the discrete topology ι , for which x ∈ lim ι F implies that F = { x } ↑ ; the coarsest one is the chaotic topology o , that is, X = lim o F for each filter on X . lim (cid:87) Ξ F = (cid:84) ξ ∈ Ξ lim ξ F , and lim (cid:86) Ξ F = (cid:83) ξ ∈ Ξ lim ξ F . Mind that F need not be equal to (sequential filter) defined by that sequence. Hence, is not of the form { x k : k > n } . It can be shown that this convergence is not countably paved , that is, p ( x, Seq ν ) > ℵ foreach x . SZYMON DOLECKI D
Here, a filter F finer than an element D of a pave-ment at x , hence F converges to x . By β F we denote the set ofultrafilters that are finer than a filter F . Accordingly, F ≥ D ( F is finer than D ) if and only if β F ⊂ β D .4. From topologies to pretopologies
Pretopologies constitute a first generalization of topologies, and have alreadybeen considered, under various names, by
Sierpiński, Čech, Hausdorff , and
Choquet .On one hand, pretopologies include topologies, which has been so far the mostknown and studied class of convergences. They have many similarities with topolo-gies, and one difference: the adherence , an analogue of topological closure , is ingeneral not idempotent. On the other hand, the class of pretopologies has a muchsimpler structure than the class of topologies.A convergence is a pretopology if for each point (of the underlying set), there is acoarsest filter converging to that point. Therefore, a convergence is a pretopologyif and only if its paving number is .For an arbitrary convergence θ , the filter V θ ( x ) := (cid:92) {F : x ∈ lim θ F} is called the vicinity filter of θ at x . Proposition.
A convergence ξ is a pretopology if and only if x ∈ lim ξ V ξ ( x ) foreach x in the underlying set | ξ | . Consequently, each topology ξ is a pretopology, and if it is, then V ξ ( x ) = N ξ ( x ) is the neighborhood filter of ξ at x .If A is a family of subsets of X , then the grill of A is defined by A := (cid:92) A ∈A { H ⊂ X : A ∩ H (cid:54) = ∅ } . For an arbitrary convergence θ , the adherence adh θ A of a set A can be definedby(set-adherence) x ∈ adh θ A ⇐⇒ A ∈ V θ ( x ) . For any convergence θ , a set A is called θ -closed if adh θ A ⊂ A . The θ -closure isdefined by cl θ A := (cid:84) H ⊃ A { H : adh θ H ⊂ H } .Notice that if ξ is a pretopology, then by (set-adherence) its adherence determinesall its vicinity filter, hence its convergent filters. Proposition.
A pretopology is a topology if and only if its adherence is idempotent.
CQUIRING A DIMENSION: FROM TOPOLOGY TO CONVERGENCE THEORY 5
In other terms, a pretopology ξ is topological whenever adh ξ (adh ξ A ) ⊂ adh ξ A for each A . In this case, adh ξ A is, of course, ξ -closed and and equal to cl ξ A , theclosure of A . Topologies, pretopologies , and many other fundamental classes of convergencesare projective . This means that for each convergence θ , there exists a finest topology Jθ among the topologies that are coarser than θ . It is easy to see that so defined J is concrete ( | Jθ | = | θ | ), increasing ( θ ≤ θ implies Jθ ≤ Jθ ), idempotent ( J ( Jθ ) = Jθ ), as well as descending ( Jθ ≤ θ ) ( ). If J preserves continuity, thatis, if C ( ξ, τ ) ⊂ C ( Jξ, Jτ ) for any convergences ξ and τ , then J is a concrete functor ( ). Definition.
A concrete, increasing, idempotent and descending functor J is calleda ( concrete ) reflector. Then the class of convergences ξ fulfilling Jξ = ξ is called reflective .Topologies an pretopologies are reflective; the reflector T on the class of topolo-gies is called the topologizer , the reflector S on the class of pretopologies is calledthe pretopologizer . Both admit similar explicit descriptions lim T θ F = (cid:92) H ∈F cl θ H, lim S θ F = (cid:92) H ∈F adh θ H, The objectwise use of functors associated with various classes of convergences,constitutes a sort of calculus, enabling to perceive in a unified way miscellaneousaspects of convergences, hence in particular of topologies.5.
Adherence-determined classes of convergences
Let ξ be a convergence on X , and let A be a family of subsets of X . The adherence adh ξ A is defined by(adherence) adh ξ A := (cid:91) H⊂A lim ξ H . In particular, if A ⊂ X , then adh ξ A = adh ξ { A } , which is the set-adherence , alreadyintroduced in (set-adherence). It is straightforward that, for each filter F on X , adh ξ F = (cid:91) U∈ β F lim ξ U . We denote by F the class of all filters, by F the class of countably based filters, andby F the class of principal (or finitely based ) filters. By convention, if F ⊂ H ⊂ F then H X is the set of filters on X that belong to the class H ( ).Let H be a class of filters. We say that H is initial if f − [ H ] ∈ H for each H ∈ H ,final if f [ H ] ∈ H for each H ∈ H ( ). for each convergences θ, θ , θ Basic facts from category theory are used here instrumentally, so to say, objectwise. Functorsare certain maps defined on classes of morphisms, and then specialized to the classes of objectsviewed as identity morphisms. Because the category of convergences with continuous maps asmorphisms is concrete over the category of sets, it is enough to define concrete functors merelyon objects. In particular, the filters from F X are of the form A ↑ := { F ⊂ X : A ⊂ F } for some A ⊂ X . Of course, it is understood that if f ∈ Y X and H ∈ H Y then f − [ H ] ∈ H X , and if H ∈ H X then f [ H ] ∈ H Y . SZYMON DOLECKI
Assume that H is an initial class of filters. Then a convergence ξ is called H -adherence-determined if lim ξ F ⊃ (cid:84) H (cid:51)H⊂F adh ξ H . The class of H - adherence-determined convergences is concretely reflective , and the reflector A H fulfills lim A H ξ F = (cid:92) H (cid:51)H⊂F adh ξ H , that is, x ∈ lim A H ξ F provided that x ∈ adh ξ H for each H ∈ H such that H ⊂ F . Topologies
Hypotopologies
Paratopologies
Pretopologies
Pseudotopologies
Figure 5.1.
Fundamental classes of adherence-determined convergencesSince we assume that F ⊂ H ⊂ F , the F -adherence-determined pretopologies form the narrowest class, and F -adherence-determined pseudotopologies the largest.By the way, topologies are not adherence-determined.We shall yet consider two intermediate classes: paratopologies , corresponding tofor which S = A F , and hypotopologies , corresponding to the class F ∧ of count-ably deep filters ( ), for which S ∧ = A F ∧ . It was recently observed [16] thatpretopologies constitute the intersection of paratopologies an hypotopologies.By the way, as all functors, the adherence-determined reflectors preserve conti-nuity, but moreover fulfill(5.1) A H ( f − τ ) = f − ( A H τ ) for each f and τ . In particular, if f is an injection, then (5.1) means that A H commutes with the construction of subspaces for F ⊂ H ⊂ F ! Mind that the topologizer T is not of the form A H , and does not commute with the constructionof subspaces. 6. Pseudotopologies
We have seen that pseudotopologies constitute an adherence-determined classwith respect to the class F of all filters. It easily follows from the definition that if ξ is a pseudotopology, then(pseudotopologizer) lim S ξ F = (cid:92) U∈ β F lim ξ U , That is, the filters F such that F ⊂ F and card F is countable, then (cid:84) F ∈ F . CQUIRING A DIMENSION: FROM TOPOLOGY TO CONVERGENCE THEORY 7 where β F stands for the set of ultrafilters that are finer than a filter F . It isremarkable that the pseudotopologizer S commutes with arbitrary products, thatis, if Ξ is a set of convergences, then(commutation) S( (cid:89) Ξ) = (cid:89) ξ ∈ Ξ S ξ. This is because S commutes with construction of initial convergence (as an adherence-determined reflector), and also with arbitrary suprema ( ).Although F ( ξ × ξ ) ≥ F ξ × F ξ holds for each functor F , the converse is ratheran exception. For instance, the pretopologizer S commutes with the constructionof initial convergence (like the pseudotopologizer S ), but not with suprema, hencenot with products. 7. Quotient maps
A first example is that of quotient maps. In topology, a continuous map f ∈ C ( ξ, τ ) (between topologies ξ and τ ) is said to be quotient if τ ≥ T( f ξ ) , hence,because of the continuity assumption, τ = T( f ξ ) . Let us remark that, if ξ is atopology, that is, T ξ = ξ , then the final convergence f ξ need not be a topology;actually, it can be, so to say, almost anything, as each finitely deep convergence ( )is a convergence quotient of topologies.It has long been known that quotient maps preserve some properties, like se-quentiality , but do not preserve others, like Fréchetness . For this reason, numerousquotient-like maps ( quotient, hereditarily quotient, countably biquotient, biquotient,triquotient, almost open ) and their preservation properties were intensively inves-tigated. In his [11],
E. Michael gathered, generalized, and refined numerouspreservation existent results (A. V. Arhangel’skii [1],
V. I. Ponomarev [17],
S. Hanai [9],
F. Siwiec [18], and others) for these quotient-like maps. Richnessand complexity of these investigations made of this quotient quest a veritable jungle( ).Using convergence-theoretic methods [3], it was possible to figure out that vir-tually all these quotient-like maps follow the same pattern, namely they are of theform( J -quotient map) τ ≥ J ( f ξ ) , where J is a reflector on a subclass of convergences, for a map f ∈ C ( ξ, τ ) ( ), apanorama that was unavailable within the framework of topologies. In particular,a map f fulfilling ( J -quotient map) is quotient if J = T , hereditarily quotient if Let us first prove that S( (cid:87) Ξ) = (cid:87) ξ ∈ Ξ S ξ. Indeed, lim S( (cid:87) Ξ) F = (cid:84) U∈ β F lim (cid:87) Ξ U ,which is equal to = (cid:84) U∈ β F (cid:84) ξ ∈ Ξ lim ξ U . By commuting the intersections, we get (cid:84) ξ ∈ Ξ (cid:84) U∈ β F lim ξ U = (cid:84) ξ ∈ Ξ lim S ξ U = lim (cid:87) ξ ∈ Ξ S ξ F .By the definition of product, (cid:81) Ξ = (cid:87) ξ ∈ Ξ p − ξ ξ , where p ξ : (cid:81) ζ ∈ Ξ | ζ | −→ | ξ | is the ξ -projection.Therefore, by (5.1), S( (cid:81) Ξ) = S( (cid:87) ξ ∈ Ξ p − ξ ξ ) = (cid:87) ξ ∈ Ξ S( p − ξ ξ ) = (cid:87) ξ ∈ Ξ p − ξ (S ξ ) = (cid:81) ξ ∈ Ξ S ξ . A convergence θ is called finitely deep if lim θ F ∩ lim θ F ⊂ lim θ ( F ∩ F ) for any F and F . A metaphor came, when I tackled to decorticate this article. I realized that I would not graspits underlying ideas, unless I transform the jungle into an Italian garden. I evoked it during aconference in honor of
Peter Collins and
Mike Reed in Oxford in 2006, and
Ernest Michael ,who attended, appreciated. In fact, in various problems the continuity of a quotient-like map is inessential, and can bedropped.
SZYMON DOLECKI
Figure 7.1.
A multiple quotient quest: transforming a jungle intoan Italian garden. J = S (pretopologizer), countably biquotient if J = S ( paratopologizer ), biquotient if J = S ( pseudotopologizer ), and almost open if J = I ( identity functor ). By theway, it is often handy to say H -quotient instead of A H -quotient .Biquotient maps are the only among the listed classes that are preserved byarbitrary products, which, of course, is due to (commutation).Of course, ( J -quotient map) transcends topologies, but when limited to topolo-gies ξ and τ , it yields a topological conclusion, having passed beyond. The name hereditarily quotient, traditonally used in the topological context, is due to the factthat each restriction of a S -quotient map remains a S -quotient ( ).It turns out that sundry properties, like sequentiality, Fréchetness , local com-pactness , and so on, appear as solutions θ of functorial inequalities of the type( JE -property) θ ≥ JEθ, where J is a concrete reflector, and E is an appropriate concrete coreflector , that isa concrete, increasing, idempotent and ascending ( θ ≤ Eθ ) functor (compare withthe definition of concrete reflector ). Example.
A topology is called sequential if each sequentially closed set is closed.If ξ is a topology, then Seq ξ is the coarsest sequential convergence, in general non-topological, that is finer than ξ . Then T Seq ξ stands for the topology, for whichthe open sets and the closed sets are determined by sequential filters, that is, aresequentially open and closed, respectively. Therefore, a topology ξ is sequential ifit coincides with T Seq ξ , which is equivalent to ξ ≥ T Seq ξ. A convergence ξ is called Fréchet if x ∈ adh ξ A implies the existence of a sequentialfilter E such that A ∈ E and x ∈ lim ξ E . It is straightforward that ξ is Fréchetwhenever ξ ≥ S Seq ξ, where S is the pretopologizer. Indeed, let ξ be a convergence on X , and τ be a convergence on Y . If f ∈ Y X fulfills τ ≥ S ( fξ ) , then for B ⊂ Y and the injection j B ∈ Y B , j − B τ ≥ j − B S ( fξ ) = S ( j − B ◦ f ) ξ, and the final convergence of ξ by j − B ◦ f is equal to the final convergence of j − f − ( B ) ξ by j − B ◦ f . CQUIRING A DIMENSION: FROM TOPOLOGY TO CONVERGENCE THEORY 9
Now a preservation scheme becomes manifest ( ). Theorem. If ξ has ( JE -property ), and f ∈ C ( ξ, τ ) is a ( J -quotient map ) , then τ has ( JE -property ) . Let us illustrate the preservation result above by the two properties discussedin the example. A more exhaustive list of special cases of this theorem, can befound in [3], where all of 20 entries correspond to theorems, many of which weredemonstrated in numerous papers. See also [8, p. 400].
Corollary.
A continuous quotient of a sequential topology is sequential. A contin-uous hereditarily quotient of a Fréchet topology is Fréchet. Exponential reflective classes
Definition.
A class J is called exponential if [ ξ, σ ] ∈ J for any convergence ξ ,provided that σ ∈ J , whereThe σ - dual convergence [ ξ, σ ] of ξ is the coarsest convergence on C ( ξ, σ ) , forwhich the evaluation map ev( x, f ) = (cid:104) x, f (cid:105) := f ( x ) is jointly continuous, that is, ev ∈ C ( ξ × [ ξ, σ ] , σ ) , that is, ξ × [ ξ, σ ] ≥ ev − σ. Theorem.
A reflector is exponential if and only if it commutes with finite products.
Here is a simple proof of sufficiency ( ).We understand now why the Kuratowski convergence on the hyperspaces ofclosed sets, considered by Choquet in [2], is not topological ( ).Given any reflector J , there exists the least exponential reflector Epi J such that J ≤ Epi J ( ). The corresponding least exponential reflective class fix(Epi J ) in-cluding fix( J ) , is called the exponential hull of J . Duality theory, developed by F.Mynard [12, 13], and others, allows to characterize exponential hulls.A construction uses the σ -dual convergence [[ ξ, σ ] , σ ] of the σ -dual convergence [ ξ, σ ] of ξ , which is a convergence on C ([ ξ, σ ] , σ ) . Then Epi σ ξ = j − [[ ξ, σ ] , σ ] , that Let ξ ≥ JEξ , ( J -quotient map) and f ∈ C ( ξ, τ ) . Then fξ ≥ f ( JEξ ) ≥ JE ( fξ ) , the lastinequality being consequence of f ( F ξ ) ≥ F ( fξ ), valid for each functor F . By ( J -quotient map)and idempotency of J , we infer τ ≥ J ( fξ ) ≥ JE ( fξ ) ≥ JEτ , the last inequality entailed bycontinuity: fξ ≥ τ . As a result, τ ≥ JEτ . Let σ = Jσ . By definition,(duality) ξ × [ ξ, σ ] ≥ ev − σ, and [ ξ, σ ] is the coarsest convergence, for which the inequality above holds. If J commutes withfinite products then, from (duality), ξ × J [ ξ, σ ] ≥ Jξ × J [ ξ, σ ] ≥ J ( ξ × [ ξ, σ ]) ≥ J (ev − σ ) ≥ ev − ( Jσ ) = ev − σ, the last inequality following from F ( fσ ) ≥ f − ( F σ ) , valid for each functor F . As, by assumption, [ ξ, σ ] is the coarsest convergence fulfilling (duality), J [ ξ, σ ] ≥ [ ξ, σ ] , hence J [ ξ, σ ] = [ ξ, σ ] , because J is a reflector. Indeed, if τ is a topology on a set X , then the (upper) Kuratowski convergence on thehyperspace consisting of all τ -closed sets is [ τ, $] , where the Sierpiński topology $ on { , } , theclosed sets of which are ∅ , { } , and { , } . The hyperspace can be identified with C ( τ, $) , thespace of continuous from τ to $ . Accordingly, A is τ -closed if and only if the characteristic function χ A ∈ { , } X , that is, A = { x ∈ X : χ A ( x ) = 1 } , fulfills χ A ∈ C ( τ, $) . It follows that J is exponential if and only if J = Epi J . is, the initial convergence of the σ -bidual convergence by the natural injection j : X −→ Z ( Z X ) , which turns out to be continuous: j ∈ C ( ξ, [[ ξ, σ ] , σ ]) . Finally, Epi J ξ := (cid:95) σ = Jσ Epi σ ξ. It turns out that the two most important non-topological convergences, introducedby G.
Choquet, are intimately related by duality.
Theorem.
The exponential hull of the class of pretopologies is the class of pseu-dotopologies.
See also ( ). 9. Compactness versus cover compactness
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 pointin A , or else, each filter on A has an adherence point in A . Many authors requirethat, besides, the topology be Hausdorff.A convergence is, in general, not determined by its open sets, and thus opencovers are not an adequate concept in this context. A natural extension to conver-gence spaces of the notion of cover is used to define cover-compact sets. The pointis that cover-compactness and filter-compactness are no longer equivalent for gen-eral convergences. Moreover, it turns out that cover-compactness is not preservedunder continuous maps. Definition.
Let ξ be a convergence on X . A family P of subsets of X is called a ξ -cover of a set A , if P ∩ F (cid:54) = ∅ for every filter F such that A ∩ lim ξ F (cid:54) = ∅ .Specializing the definition above to a topology ξ on a set X , we infer that P isa ξ -cover of A if and only if A ⊂ (cid:83) P ∈P inh ξ P , where inh ξ P := X \ adh ξ ( X \ A ) isthe ξ -inherence of P .Endowed with this extended concept of cover, we are in a position to discusscover-compactness for general convergences. Definition.
A set A is said to be ξ - cover-compact if for each ξ -cover of A , thereexists a finite ξ -subcover of A ; ξ -compact if, for each filter H ,(compact set) A ∈ H = ⇒ adh ξ H ∩ A (cid:54) = ∅ . The following simple ( ), but very consequential observation [4] enables to easilycompare the two variants. Proposition.
A family P is a ξ -cover of A if and only if adh ξ P c ∩ A = ∅ . Let us mention that
Epi S ξ = S ξ = j − [[ ξ, (cid:85) ] , (cid:85) ] , where (cid:85) is the Bourdaud pretopology . The
Bourdaud pretopology (cid:85) is defined on { , , } by convergence of ultrafilters as follows lim (cid:85) { } ↑ = { , } , lim ¥ { } ↑ = { , , } , lim ¥ { } ↑ = { , , } . The exponential hull of topologies is the class of epitopologies , defined by
P. Antoine , and then
Epi T ξ = j − [[ ξ, $] , $] . A family P is not a ξ -cover of a set A , whenever there exists a filter F such that A ∩ lim ξ F (cid:54) = ∅ and P / ∈ F for each P ∈ P . In other words, F ∩ P c = F \ P (cid:54) = ∅ for each P ∈ P and each F ∈ F ,equivalently, P c F , that is, A ∩ adh ξ P c (cid:54) = ∅ . CQUIRING A DIMENSION: FROM TOPOLOGY TO CONVERGENCE THEORY 11
To this end, we focus on ideal covers. A family of subsets of a given set is calledan ideal if ( P ∈ P ) ∧ ( P ∈ P ) ⇐⇒ P ∪ P ∈ P . Clearly, P is an ideal of subsets of X if and only if P c is a filter on X . Passingfrom arbitrary covers to ideal covers makes no difference in topology, but doesmake in general. By the preceding proposition, on setting H = P c , we characterizefilter-compactness in terms of ideal covers: Proposition.
A set A is ξ -compact if and only if A ∈ P for each ideal ξ -cover P of A . Cover-compactness implies (filter)-compactness for pretopologies. Indeed, if ξ isa pretopology, and A is ξ -cover-compact, then in particular, for each ideal ξ -cover P of A , there exists a finite P ⊂ P such that A ⊂ (cid:83) P ∈P inh ξ P ⊂ inh ξ (cid:83) P ∈P P ⊂ (cid:83) P ∈P P ∈ P , because P is an ideal. Hence A ∈ P , so that A is ξ -compact.On the other hand, there exist pretopologies, where the two notions differ [8,Example IX.11.8].Moreover, each finite set is ξ -compact for any convergence ξ ( ), but Proposition ([14, 8]) . A pseudotopology, the finite subsets of which are cover-compact, is a pretopology.Proof. If ξ is not a pretopology, then there is x ∈ | ξ | such that each ξ -pavementof x is infinite. Thus if Q is a ξ -cover of { x } and P is a ξ -pavement at { x } , then Q ∩ P (cid:54) = ∅ for each P ∈ P , so that Q cannot be finite. (cid:3) Corollary.
Continuous maps between non-pretopological spaces do not preservecover-compactness ( ) . Extensions of the concept of compactness
Compact families of sets generalize both compact sets and convergent filters, andthis generalization is not just a whim. It has important applications, and, perhapsmore importantly, evidences mathematical laws that remained invisible on the levelof compactness of sets.Let ξ be a convergence on X . A family A of subsets of X is said to be ξ -compactat a family B of subsets of X if, for each filter H ,(compact family) A ⊂ H = ⇒ adh ξ H ∈ B . In particular, A is called ξ -compact if it is ξ -compact at itself; ξ -compactoid if it is ξ -compact at X ( ).It is clear that a subset A of X is ξ - compact ( ξ - compactoid ), whenever A ↑ := { F ⊂ X : A ⊂ F } is ( ). On the other hand, it is straightforward that F is In fact, if a filter H fulfills { x } ∈ H then x ∈ (cid:84) H , and thus x ∈ adh ξ H , equivalently { x } ∩ adh ξ H (cid:54) = ∅ . If ξ is a convergence on X such that p ( x , ξ ) is infinite, ι is the discrete topology on X , thenfor the identity map i X ∈ C ( ι, ξ ) , the image i ( { x } ) is not ξ -cover-compact, but { x } is ι -covercompact. The set κ ( ξ ) of all ξ -compact (isotone) families on X = | ξ | fulfills: ∅ , X ∈ κ ( ξ ) , {A j : j ∈ J } ⊂ κ ( ξ ) entails (cid:83) j ∈ J A j ∈ κ ( ξ ) , and (cid:84) j ∈ J A j ∈ κ ( ξ ) , whenever J is finite. In other words, κ ( ξ ) has the properties of a family of open sets of a topology on X . ξ -compact at { x } if and only if x ∈ lim S ξ F . Incidentally, it is straightforward that ξ -compactness and S ξ -compactness coincide.This simple fact prefigures the pseudotopological nature of compactness, whichwill be evidenced in a moment.At this point, it will be instrumental to consider again the notion of grill, froma somewhat different perspective. Recall that A := (cid:84) A ∈A { H ⊂ X : A ∩ H (cid:54) = ∅ } for a family A of subsets of X . Now, for another family H on X , the condition H ⊂ A is equivalent to
A ⊂ H , so we denote this relation symmetrically, by H A ( ). If A is on X , and B is on Y , and f : X −→ Y , then it is easy to seethat(grill) f [ A ] B ⇐⇒ A f − [ B ] . For a given convergence ξ on X , define the associated characteristic convergence χ ξ by(characteristic) lim χ ξ F := (cid:40) X lim ξ F (cid:54) = ∅ , ∅ lim ξ F = ∅ . It is immediate that, for a set Ξ of convergences,(characteristic of product) χ (cid:81) Ξ = (cid:89) ξ ∈ Ξ χ ξ . Lemma.
A filter F is ξ -compactoid if and only if lim S χ ξ F (cid:54) = ∅ .Proof. By (pseudotopologizer), lim S χ ξ F (cid:54) = ∅ if and only if lim χ ξ U (cid:54) = ∅ for each U ∈ β F , equivalently, by (characteristic), lim ξ U (cid:54) = ∅ for each U ∈ β F . (cid:3) As an immediate consequence of this lemma and of (commutation),
Theorem (Generalized Tikhonov Theorem) . A filter F is (cid:81) Ξ -compactoid if andonly if p ξ [ F ] is ξ -compactoid for each ξ ∈ Ξ .Proof. By (characteristic of product) and (commutation), S( χ (cid:81) Ξ ) = S( (cid:81) ξ ∈ Ξ χ ξ ) = (cid:81) ξ ∈ Ξ S χ ξ . The proof is complete in virtue of Lemma above. (cid:3) If we restrict filters H in (compact family) to a class H of filters, then we obtaina notion of H -compactness . We assume that F ⊂ H ⊂ F , that is, that the saidclass includes all principal filters. A is said to be ξ - H -compact at B if ∀ H∈ H A ⊂ H = ⇒ adh ξ H ∈ B . Some instances of this notion have been already known in topological context,like F -compactness , that is, countable compactness ( ), or F ∧ -compactness , thatis, Lindelöf property. If H = A H is the H -adherence-determined reflector, then afilter F is H - compactoid for ξ , whenever ( ) lim Hχ ξ F (cid:54) = ∅ . Of course, once established for special reflectors, the formula above can be used asdefinition of H -compactness for arbitrary reflectors H .Observe that, for other refectors H than the pseudotopologizer, H -compactnessis nor preserved even by finite products if H does not commute with such products. Of course, H A whenever H ∩ A (cid:54) = ∅ for each H ∈ H and A ∈ A . By the way, sequential compactness of ξ coincides with F -compactness of Seq ξ . Recall that x ∈ lim Hξ F whenever F is ξ - H -compact at { x } . CQUIRING A DIMENSION: FROM TOPOLOGY TO CONVERGENCE THEORY 13
Perfect-like maps
A step further is to extend H -compactness to relations. Roughly speaking ( ),a relation R is H - compact if y ∈ lim F implies that R [ F ] is H -compact at Ry . Con-tinuous maps and various quotient maps can be characterized in terms of compactrelations [14]. However, most advantageous applications of compact relations areto various perfect-like maps.A surjective map f is H -perfect if and only if the relation f − is H -compact.For instance, F - perfect maps are precisely perfect maps are close maps withcompact fibers. F - perfect maps, or countably perfect maps are close maps withcountably compact fibers. Proposition.
A surjective map f ∈ C ( ξ, τ ) is H -perfect if and only if f − τ ≥ A H ( χ ξ ) . In particular, arbitrary product of F -perfect maps is F -perfect. This is becausethe pseudotopologizer S = A F commutes with arbitrary products, or else becausethe product of compact fiber relations is compact by the Generalized TikhonovTheorem .Perfect-like and quotient-like properties embody various degrees of converse sta-bility of maps f , or in other terms, of stability of fiber relations f − , which is theinverse relation of f . Let us rewrite these properties in expanded form, where f : | ξ | −→ | τ | .A surjective map f is H -quotient if and only if(11.1) f − (adh τ H ) ⊂ adh ξ f − [ H ] holds for each H ∈ H . A surjective map f is G -perfect if and only if(11.2) adh τ f [ G ] ⊂ f (adh ξ G ) holds for each G ∈ G . Lemma.
Let F ⊂ A ⊂ F be such that A ∈ A implies f − [ A ] ∈ A and f [ A ] ∈ A .Then every A -perfect map is A -quotient.Proof. Set G := f − [ H ] and apply f − to both sides of (11.2). As f [ f − [ H ]] = H ,because f is surjective, and since by f − ( f ( H )) ⊂ H for each H , we get f − (adh τ H ) ⊂ f − (adh τ f [ f − [ H ]]) ⊂ f − ( f (adh ξ f − [ H ])) ⊂ adh ξ f − [ H ] . (cid:3) Let θ be a convergence on W and σ be a convergence on Z . A relation R is called J -compact if w ∈ lim θ F implies that R ( w ) σ H for each H R [ F ] such that H ∈ J . perfect-like ⇒ quotient-like reflectoropenalmost open identity I perfect ⇒ biquotient pseudotopologizer S countably perfect ⇒ countably biquotient paratopologizer S adherent ⇒ hereditarily quotient pretopologizer S closed ⇒ topologically quotient topologizer T Table 1.
Interrelations between perfect-like maps, quotient-likemaps and reflective classes.No arrow can be reversed. Indeed,
Example.
Let f : R → S be given by f ( x ) := (cos 2 πx, sin 2 πx ) . It followsimmediately from the proposition above that f is open, hence, has all the propertiesfrom the right-hand column. Notice that the the set { n + n : n ∈ N } is closed, butits image by f is not closed, so that f is not closed, and thus has no property fromthe left-hand column. 12. Conclusions
I hope that these outlines allow to grasp the essence of convergence theory. Sureenough, only few aspects have been touched upon, and most remain beyond thispresentation.For example, various types of compactness are instances of numerous kindsof completeness. The completeness number compl( ξ ) of a convergence ξ is theleast cardinality of a collection of ξ -non-adherent filters that fill the set of ξ -non-convergent ultrafilters in the Stone space. This way, compact convergences ξ arecharacterized by compl( ξ ) = 0 , locally compactoid by compl( ξ ) < ∞ , and topologi-cally complete by compl( ξ ) ≤ ℵ . Each convergence has its completeness number;for the “very incomplete” space of rational numbers , this number is the dominatingnumbe r d .It was shown in this paper that a generalization of Tikhonov Theorem is a simplecorollary of the commutation of the pseudotopologizer with arbitrary products. Itturns out that it is also a simple consequence of a theorem on the completenessnumber of products [6][8].It appears that compl( ξ ) is equal to the (free) pseudo-paving number of the dualconvergence [ ξ, $] at ∅ , and the (free) paving number of [ ξ, $] at ∅ is equal to theultra-completeness number of ξ [15].We see that, in the framework of topologies, it would be impossible to considera property dual to Čech completeness (countable completeness number), becausethe paving and pseudo-paving numbers of a topology do not exceed .I could long display similar examples, but I expect that these few exhibited inthis paper would convince you of the interest of convergence theory. References [1] A. V. Arhangel’skii. Some types of factor mappings and the relations between classes oftopological spaces.
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