# Subobjects and Compactness in Point-Free Convergence

Authors: Emilio Angulo; Frédéric Mynard
Journal: Journal of Mathematics
 (2023)

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## Abstract

We consider subobjects in the context of point-free convergence (in the sense of Goubault-Larrecq and Mynard), characterizing extremal monomorphisms in the opposite category of that of convergence lattices. It turns out that special ones are needed to capture the notion of subspace. We call them standard and they essentially depend on one element of the convergence lattice. We introduce notions of compactness and closedness for general filters on a convergence lattice, obtaining adequate notions for standard extremal monos by restricting ourselves to principal filters. The classical facts that a closed subset of a compact space is compact and that a compact subspace of a Hausdorff space is closed find generalizations in the point-free setting under the form of general statements about filters. We also give a point-free analog of the classical fact that a continuous bijection from a compact pseudotopology to a Hausdorff pseudotopology is a homeomorphism.


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## Body

## 1. Introduction

Recall that a convergenceξ on a set X is a relation between the set FPX of (set-theoretic) filters on X and the set X, denoted by(1)x∈limξℱ,if ℱ and x are ξ-related, subject to the following two axioms:(2)monotoneℱ⊂G⇒limξℱ⊂limξG,pointaxiomx∈limξx↑,for every x∈X and every ℱ,G∈FPX. Continuity of a map f:X,ξ⟶Y,τ is simply preservation of limits, that is,(3)continuityflimξℱ⊂limτfℱ,where fℱ=B⊂Y:f−1B∈ℱ∈FPY is the image filter of ℱ under f.LetConv denote the category of convergence spaces and continuous maps. This is a topological category in which subspaces are defined as usual as the initial structure for the inclusion map; namely, if A⊂X and X,τ is a convergence space, then the induced convergenceτ|A on A is defined by limτ|Aℱ=limτℱ↑X∩A, where ℱ↑X is the filter generated on X by the filter ℱ on A.The categoryTop of topological spaces and continuous maps is a concretely reflective subcategory of Conv. See e.g., [1], for a systematic treatment of Conv and of classical topology from that viewpoint.In [2], Goubault-Larrecq and Mynard introduce a point-free generalization of Conv in which the function(4)limξ:FℙX→ℙX,is abstracted away to a monotone function(5)lim:FL⟶L,from (order-theoretic) filters on a lattice L to L. Note that (point axiom) is not part of the axiomatic in this point-free version of convergence spaces, though the notion can also be recovered (as so-called centered convergence lattices) in an abstract order-theoretic form.In the context of this paper, a lattice is an ordered set with all finite joins and meets, including the empty ones, so that our lattices have a greatest element, usually denoted by⊤ and a least element usually denoted by ⊥. Lattice morphisms preserve all finite joins and meets including the empty ones, so that our lattice morphisms send the greatest element to greatest element and the least to least. Let Lat denote the corresponding category. By a category of lattices, we mean a subcategory of Lat.Definition 1.
Given a categoryC of lattices, a convergenceC-objectL,lim is a C-object L together with a monotone map lim:FL⟶L. The objects of the category CConv are the convergence C-objects, and the morphisms φ:L⟶L′ are the C-morphisms that are continuous in the sense that for everyℱ∈FL′,(6)ptfree continuitylimL′ℱ≤φlimLφ−1ℱ,where φ−1ℱ=ℓ∈L:φℓ∈ℱ. The morphism φ is final (in the usual categorical sense) if we have equality in (ptfree continuity).
The categoryConv embeds coreflectively into CConvop when C is the category Frm of frames or CFrm of coframes: the powerset-functor ℙ:Conv⟶CConvop sending X,ξ to ℙX,limξ and f:X,ξ⟶Y,τ to ℙf=f−1:ℙY⟶ℙX (in CConv) is then right-adjoint to the point-functor pt:CConvop⟶Conv (the coreflector), where the underlying set of ptL, the set of “points” of L, is the set of CConv-morphisms from L to ℙ1, hence depending on the choice of C. The convergence structure on ptL is given by(7)Conv on ptlimptLℱ=limLℱ°•,where ℓ•=φ∈ptL:φℓ=1 and ℱ°=ℓ∈L:ℓ•∈ℱ. Finally, if φ∈CConvL,L′, then ptφ:ptL′⟶ptL is defined by ptφf=f∘φ.
In the same work [2], adjunctions between Conv, CConv, and some of their important subcategories are proved, and it is also shown that, if C is the category of coframes, CConv is a topological category. Though the point-free analog of the category of topological spaces in CConv is not related in a straightforward way with the classical approach to point-free topology [2], ([2], section 8.5) shows how the latter can be recovered, realizing the category of locales as a reflective subcategory of the opposite of the category of strong topological coframes in the CConv context. This new framework already proved further versatility, as the category of convergence approach spaces in the sense of [3] can also be faithfully represented in CConv [4].
In the present work, we explore the basic concepts of subspace, compactness, closedness, and Hausdorffness in the setting ofCConv. More specifically, we characterize the extremal monomorphisms of CConvop, that is, the extremal epimorphism of CConv, as the C-extremal epimorphism that are also final in CConv. Though this provides an adequate notion of subobject, a stricter notion is necessary to capture exactly those subobjects of the convergence lattice ℙX,limξ coming from a convergence space X,ξ that represent a subspace of X,ξ. We call such extremal epis standard. They essentially depend on a single element of the convergence lattice, like picking a subspace of X depends on picking an element of ℙX.
In the context of convergence spaces, the notions of closedness and compactness have been generalized to families of subsets rather than single subsets, e.g., [1, 5, 6] and references therein, to the effect that a subset has the property if and only if its principal filter does. Similarly, in the point-free context, we consider compactness and closedness for filters, and the case of a principal filter provides notions for standard extremal epis. We also consider a second direction of generalization, defining compactness for morphisms, though few results are obtained in this direction. This avenue may be explored further in future work.
Throughout this work, we are going to use definitions and notations from [2] for convergence lattices, from [7, 8] for categorical notions, and from [9, 10] for lattice theory and locales. Moreover, though we work with categories C of lattices, some notions involve infinite suprema or infima. Whenever they appear, the existence of such infima or suprema is implicitly assumed, so that the notion is most naturally considered within convergence lattices that are complete lattices, though the morphisms remain those of CConv.


## 2. Subobjects in Categories of Convergence Lattices

As pointed out in ([10], III. 1) and ([8], 1.2.2.6 2), the extremal monomorphisms are the categorical candidates to represent the subobjects of a topological category. Because we are working in CConvop, the extremal monomorphisms there are the extremal epimorphisms in CConv. Recall that an epimorphism e is extremal if whenever e=m∘f for a morphism f and a monomorphism m, m is an isomorphism.It turns out that these extremal epimorphisms have the expected topological behavior of being final.Theorem 1.
LetC be a category of lattices. The extremal epimorphisms of CConv are exactly extremal epimorphisms of C that are final in CConv.Proof.
Assume thatφ:L,limL⟶L′,limL′ is a CConv-extremal epimorphism and let m∘f be a C-factorization of φ, where m is a C-monomorphism.
Note that, even ifC is not a category of coframes, we can always construct the final structures for sources with only one morphism ([2], Corollary 3.3). Here we can give Z the final structure limZℱ=flimLf−1ℱ. As f is final and φ=m∘f is a morphism, we conclude that m is a CConv-morphism. Therefore, m is a CConv-isomorphism because φ is an extremal epimorphism in CConv. In particular, m is an isomorphism of C. Hence, φ is an extremal epimorphism of C.
Moreover,φ:L,limL⟶L′,limL′ is final. To see this, consider the next commutative triangle
Assume conversely thatφ:L,limL⟶L′,limL′ is an extremal epimorphism of C that is final in CConv. Let m∘f be a CConv factorization of φ with m a CConv-monomorphism:
Thenm is in particular a C-monomorphism, hence a C-isomorphism, which means that f=m−1φ as maps, and therefore, because φ is final in CConv, we conclude that m−1 is a CConv morphism.Remark 1.
Note that whenC is the category of frames, the extremal epimorphisms of C are the onto C-morphisms ([10], Proposition 1.1.3). This is also true for the category of coframes.
Note that with this notion of subobject, there may be more subobjects of a convergence spaceX,ξ seen as a convergence lattice ℙX,limξ than subspaces of X,ξ.Proposition 1.
LetX,ξ be a convergence space. Given an extremal epimorphism φ:ℙX,limξ⟶L′ of CConv, the following are equivalent:(1)
There isA⊂X with inclusion map i:A⟶X such thatcommutes, wherej:ℙA,limξ|A⟶L′ is an isomorphism (equivalently a monomorphism) of CConv.(2)
ℓφ≔∧φ−1⊤L′ satisfies φℓφ=⊤L′ and φ is injective on ↓ℓφ.Proof.
1⟹2: note that in (1) j is an isomorphism whenever it is a monomorphism because φ is an extremal epimorphism. Assuming (1),(8)φ−1⊤L′=ℙi−1∘j−1⊤L′=ℙi−1A=B⊂X:B⊂A,because j−1⊤L′=⊤ℙA as j is an isomorphism. Hence, ∧φ−1⊤L′=A and φA=jℙiA=jA=⊤L′ and moreover, if B≠C∈↓ℓφ, that is, B≠C∈ℙA, then ℙiB=B∩A=B and ℙiC=C∩A=C are different elements of ℙA and thus their images under the isomorphism j are different elements of L′. Hence, φB≠φC.
2⟹1: let A=∧φ−1⊤L′. Consider the map j:ℙA,limξ|A⟶L′ defined by jB=φB for every B∈ℙA⊂ℙX. As φA=⊤L′ and φ is a morphism of C, so is j. Moreover, φ is injective on ↓A so that j is a monomorphism of C and for every D∈ℙX,(9)φD=φD∩X=φD∧φX=φD∧φA=φD∩A=j∘ℙiD.
Moreover,j is continuous because ℙi is final and φ=j∘ℙi is continuous.
Recall that an epimorphisme:L⟶L′ is split if it has a right-inverse, that is, if there is a morphism s:L′⟶L such that e∘s=idL′. Split epimorphisms are extremal. Note that when A is a subspace of X,ξ as in Proposition 1, then ℙi:ℙX,limξ⟶ℙA,limξ|A is also a split epimorphism, taking s=e∘j−1, where e:ℙA⟶ℙX is eB=B. However, there are split epimorphisms that do not correspond to subspaces (e.g., Example 1 below).Definition 2.
An extremal epimorphismφ:L⟶L′ of CConvis standard if ℓφ≔∧φ−1⊤L′ satisfies φℓφ=⊤L′ and φ is injective on ↓ℓφ.Proposition 2.
An extremal epimorphismφ:L⟶L′ of CConv is standard if and only if L′ is isomorphic to a sublattice of L of the form ↓ℓ for some ℓ∈L with the limit given by(10)lim↓ℓℱ=ℓ∧limL↑ℱ,for every ℱ∈F↓ℓ.Proof.
For everyℓ∈L, the map φ:L⟶↓ℓ defined by φm=m∧ℓ is a standard extremal epimorphism if lim↓ℓ is given by (10). Indeed, φ is onto and is the identity on ↓ℓ, so that ℓ=ℓφ, φℓ=ℓ=⊤↓ℓ and φ is injective on ↓ℓ.
Conversely, given a standard extremal epimorphismφ:L⟶L′ of CConv, we can show that L′ is isomorphic to ↓ℓφ with the corresponding limit given by (10).
However, not all extremal epimorphisms are standard as shown in the following.Proposition 3.
Given a convergence latticeL,limL with at least two elements and φ∈ptL, then φ:L⟶ℙ1 is a split (hence extremal) epimorphism of CConv.Proof.
Lets:ℙ1=⊤,⊥⟶L be defined by s⊥=⊥L and s⊤=⊤L. This is continuous because limℙ1⊤=⊤ so that for every proper filter ℱ∈FL, limLℱ≤slimℙ1s−1ℱ=⊤L because s−1ℱ=⊤. Hence, φ∘s=idℙ1.Example 1.
(a split epimorphism ofLatconv that is not standard). Note that if C is the category Lat of lattices and X,ξ is a convergence space, then a point φ:ℙX⟶ℙ1 of ℙX,limξ can be identified with the filter ℱ=φ−1⊤, which is prime because φ is a lattice morphism and satisfies limξℱ∈ℱ by the continuity condition. As prime filters of ℙX are ultrafilters, points of ℙX,limξ are ultrafilters U on X satisfying limξU∈U. There may be such nonprincipal ultrafilters on a convergence space. For instance, on an infinite Noetherian topological space X,ξ, all free ultrafilters are points of ℙX,limξ (see [11] for details), which are nonstandard extremal epimorphisms, for ∧φ−1⊤=∩U=∅ and φ∅=⊥.
Let us now examine the action of the functorpt on extremal epimorphism of CConv.Theorem 2.
Ifφ:L⟶L′ is a final epimorphism of CConv (in particular an extremal epimorphism), then ptφ:ptL′⟶ptL is one-to-one and initial; that is, ptL′ is (homeomorphic to) a subspace of ptL.Proof.
ptφ is one-to-one. Indeed, if f,g:L⟶ℙ1 are two points of L′ and ptφf=ptφg, that is, f∘φ=g∘φ, then f=g because φ is an epimorphism. If φ is final, then ptφ is initial. Indeed, if f:X,ξ⟶ptL′ with ptφ∘f:X,ξ⟶ptL continuous, then ptφfx∈limptLptφ∘fℱ whenever x∈limξℱ, that is, ptφfxlimLptφfℱ∘=⊤ in ℙ1, equivalently, fx∘φlimLptφfℱ∘=⊤. We want to show that fx∈limptL′fℱ, that is, fxlimL′fℱ∘=⊤ in ℙ1. Since φ is final, limL′fℱ∘=φlimLφ−1fℱ∘, so that fxlimL′fℱ∘=fx∘φlimLφ−1fℱ∘. Hence, it is enough to show that ptφfℱ∘=φ−1fℱ∘, which follows from(11)ℓ∈φ−1fℱ∘⟺φℓ∈fℱ∘⟺φℓ•∈fℱ⟺∃F∈ℱfF⊂φℓ•⟺∃F∈ℱ,∀x∈Ffxφℓ=⊤⟺∃F∈ℱ,∀x∈Fptφfxℓ=⊤⟺ℓ•∈ptφfℱ⟺ℓ∈ptφfℱ∘.


## 3. Variants of Compactness in Convergence Lattices

We shall define notions ofadherence (the adherence defined here is the natural generalization to subsets of L of what is called raw adherence and denoted by adh0 in [2]), compactness and closedness, in the point-free convergence setting. To this end, we say that two subsets A and B of a lattice Lmesh, in symbols A#B, if a∧b>⊥ for every a∈A and every b∈B. We also write A#=ℓ∈L:ℓ#A. Note that if two filters ℱ and G on L mesh, then there is the smallest filter that contains them both.Definition 3.
LetL,limL be a convergence lattice and ℱ∈FL, we define the adherence of ℱ as(12)adhLℱ=∨G∈FLG#ℱlimG.
Note that∨G∈FLG⊃ℱlimG≤adhℱ because G∈FL:G⊃ℱ⊂G∈FL:G#ℱ, and if G#ℱ there is ℋ=↑g∧f:g∈G,f∈ℱ⊃ℱ with limℋ≥limG, so that(13)adhℱ=∨G∈FLG⊃ℱlimG.Remark 2.
As is well known, under the Axiom of Choice, denotingUL the set of maximal filters on L, the set Uℱ=U∈UL:U⊃ℱ is always nonempty and thus the adherence only depends on maximal filters via:(14)adhℱ=∨U∈UℱlimU.Lemma 1.
Iff:M⟶L is a CConv-morphism, then(15)adhLℱ≤fadhMf−1ℱ,for every ℱ∈FL.Proof.
SincelimLG≤flimMf−1G for every G∈FL with G⊃ℱ,(16)adhLℱ=∨G⊃ℱlimLG≤∨G⊃ℱflimMf−1G≤f∨G⊃ℱlimMf−1G≤f∨ℋ⊃f−1ℱlimMℋ=fadhMf−1ℱ.
Abstracting from the case ofL=ℙX,limξ, we say that ℓ∈L is compact if every ℱ∈FL with ℓ∈ℱ satisfies adhLℱ∧ℓ>⊥L. We shall generalize the notion of compactness in several directions. On one hand, compactness of an element ℓ should coincide with compactness of the corresponding standard extremal epimorphism −∧ℓ:L⟶↓ℓ for an appropriate notion of compactness of φ:L⟶L′. On the other hand, compactness has been extended to families of subsets in a very useful fashion in the context of Conv (See, e.g., [1, 5, 6, 12] and references therein) and this can be extended to the point-free setting. We start with the latter, as the corresponding notions will be useful in analyzing the former concept.


### 3.1. Compactness for Filters

Definition 4.
LetL,limL be a convergence lattice and D⊂FL. A filter ℱ∈FL is D-compact atA⊂L if(17)∀D∈D,D#ℱ⟹adhD∈A#.When D=FL, we omit the prefix D. When A=ℱ we omit “at ℱ.” Hence, ℱ is compact if(18)∀ℋ∈FL,ℋ#ℱ⟹adhℋ∈ℱ#.In the case where A=⊤, we omit “at A” and add the suffix “oid.” So ℱ is D-compactoid means that(19)∀ℋ∈D,ℋ#ℱ⟹adhℋ>⊥.
A related notion is that of nearD-compactness as introduced (in the case L=ℙX) in [5]. A filter ℱ on L is nearlyD-compact atA⊂L if(20)∀D∈D,D⊃ℱ⟹adhD∈A#.
Note that with these definitions, an elementℓ∈L is compact if and only if its principal filter ↑ℓ is a compact filter. Hence, relativizations of compactness with respect to a class of filters D (yielding notions of countable compactness, Lindelöfness, etc.) or with respect to a subset A of L can be applied to a single element, identifying it with its principal filter. As a result, we say that a convergence lattice L is D-compact if ⊤ is D-compactoid, that is, adhD>⊥ for every proper filter D∈D.Remark 3.
Note that whenD=F is the class of all filters, then D-compactness and near D-compactness are equivalent. Indeed, if ℱ is nearly compact (at A) and ℋ#ℱ there is a filter G finer than ℱ and ℋ. By near compactness, adhG∈A#. Since G⊃ℋ, adhG≤adhℋ and thus adhℋ∈A#.
It is clear that the image under theℙ functor of a compact convergence space is CConv-compact (that is, ℱ=⊤ is compact).Definition 5.
LetL,limL be a convergence lattice. A filter ℱ∈FL is closed if(21)adhℱ≤∧ℱ.Remark 4.
Note that this is a generalization to filters of the notion ofclosed element of a convergence lattice L,lim as defined in [4, 13] (which is different from the notion of closed element introduced in [2], where the present notion is called quasi-closed), where ℓ∈L is closed if(22)ℓ∈ℱ⟹limℱ≤ℓ,for every ℱ∈FL. Indeed, ℓ∈L is a closed element if and only if its principal filter ↑ℓ is closed in the sense of Definition 5.
Hence, a subspaceA,ξ|A of a convergence space X,ξ is closed if and only if A is a closed element of ℙX,limξ if and only if the principal filter of A in the convergence lattice ℙX,limξ is closed in the sense of the previous definition.Theorem 3.
LetL,limL be a convergence lattice and D⊂FL, let G⊃ℱ be filters on L, where ℱ is D-compactoid and G is closed. Then, G is nearly D-compact.Proof.
LetD∈D with D⊃G. Note that adhG≥adhD. Then, D#ℱ because G⊃ℱ, and ℱ is D-compactoid, hence,(23)adhG≥adhD>⊥.
AsG is closed, ⊥<adhD≤∧G so that(24)adhD∧g=adhD>⊥,for every g∈G and the conclusion follows.
In the case whereD=FL, we obtain, in view of Remark 3:Corollary 1.
Ifℱ is compactoid, G⊃ℱ, and G is closed then G is compact.In particular, in the caseℱ=⊤ we have the following.Corollary 2.
IfL,limL is a compact convergence lattice then every closed filter on L is compact.In particular, applying this fact to principal filters, the fact that closed subspaces of a compact convergence space are closed extends to the point-free setting: every closed element of a compact convergence lattice is compact.The point-free version of the classical fact that a continuous image of a compact set is compact will not extend straightforwardly to all morphisms in an arbitrary category of convergence lattices, but we can give a version in convergence frames.To this end, note that a class of filtersD consists of a set DL⊂FL of filters on each lattice L.Definition 6.
We say that a classD of filters is admissible if given a lattice morphism f:L⟶L′, fD∈DL′ whenever D∈DL and f−1ℋ∈DL whenever ℋ∈DL′.
Note that, in particular, ifD is admissible and L′⊂L, then every G∈DL′ generates a filter of DL.
Recall that frames are pseudocomplemented; that is, everyℓ∈L has a pseudocomplementℓ∗≔∨m∈L:m∧ℓ=⊥, which satisfies ℓ∧ℓ∗=⊥. Note that if ℓ has a pseudocomplement, then ℓ∈ℱ# if and only if ℓ∗∉ℱ. In general, a lattice or frame morphism does not need to preserve pseudocomplements, though φℓ∗≤φℓ∗ whenever ℓ and φℓ have pseudcomplements. On the other hand, a morphism of Heyting algebras preserves pseudocomplements.Lemma 2.
LetL and L′ be lattices and φ:L⟶L′ be a lattice morphism. Let ℱ∈FL′ and ℓ∈L. Then,(25)φℓ∈ℱ#⟹ℓ∈φ−1ℱ#.Ifℓis pseudocomplemented andφpreserves pseudocomplements, then the converse is true.Proof.
Ifφℓ∧m≠⊥ then ℓ∧a≠⊥ for every a∈φ−1m, for φℓ∧a=φℓ∧φa=φℓ∧m≠⊥ and φ⊥L=⊥L′. In particular, if φℓ∈ℱ# then ℓ∈φ−1ℱ#.
Conversely, ifℓ∈φ−1ℱ# then ℓ∗∉φ−1ℱ, that is, φℓ∗∉ℱ. If φ respects pseudocomplements φℓ∗=φℓ∗∉ℱ equivalently, φℓ∈ℱ#.Theorem 4.
Letg:L,limL⟶H,limH be a FrmConv morphism that preserves pseudocomplements. If ℱ∈FH is compact, then g−1ℱ is compact.Note that in the classical case whereL=ℙY, H=ℙX, and g=f−1, where f:X⟶Y, then L and H are Boolean algebras, hence complemented, and g respects complements, hence pseudocomplements.Proof.
Letℓ∈L such that gℓ∈ℱ. If ℋ#g−1ℱ, then gℋ#ℱ by Lemma 2. Thus, adhgℋ#ℱ because ℱ is compact. In particular, adhgℋ∧gℓ>⊥, that is,(26)∨W#gℋlimW∧gℓ≠⊥.
SinceL is a frame, this implies that there is W#gℋ with limW∧gℓ≠⊥.
Now,limW≤glimg−1W by (ptfree continuity). Thus,(27)⊥≠limW∧gℓ≤glimg−1W∧ℓ,so that limg−1W∧ℓ≠⊥, because g is a lattice morphism, hence sends ⊥ to ⊥.
Recall that a convergence space isHausdorff if limℱ consists of at most one point for every filter ℱ on the space. The following definition gives an abstraction to the point-free setting.Definition 7.
We say a convergence latticeL,limL is Hausdorff if for every ℱ∈FL either limℱ=⊥ or limℱ is an atom of L; that is, limℱ is minimal in L\⊥.
It is straightforward that the image underℙ of any Hausdorff convergence space is a Hausdorff convergence lattice.Theorem 5.
LetL,limL be a Hausdorff convergence frame and let ℱ be a compact filter on L. Then, ℱ is closed.Proof.
We have to prove that(28)∨ℱ⊂GlimG=adhℱ≤∧ℱ.
It will be enough to prove that(29)limG≤f,for any G⊃ℱ and any f∈ℱ. For any such G and f, adhG#ℱ by compactness of ℱ. Hence,(30)∨ℋ⊃Glimℋ∧f>⊥,equivalently,(31)∨ℋ⊃Glimℋ∧f>⊥,because L is a frame. Hence,(32)∀f∈ℱ∃ℋf⊃Glimℋf∧f>⊥.
BecauseL is Hausdorff, it follows that(33)limG≤limℋf=limℋf∧f≤f.Remark 5.
The same result can be obtained without using the assumption thatL be a frame, using maximal filters as in Remark 2, hence using the Axiom of Choice.


### 3.2. Pseudotopological Convergence Lattices and Minimality of Compact Hausdorff Structures

We call a convergence latticepseudotopological, if for every ℱ∈FL,(34)limℱ≥∧ℋ#ℱadhℋ.Note that the reverse inequality in (34) is always true (of course, (34) implicitly assumes that the infimum involved exists. Hence, pseudotopological convergence lattices are most naturally considered in the context of complete lattices).Given a convergence latticeL,limL, we define its pseudotopological modification limLS:FL⟶L by(35)limLSℱ≔∧ℋ#ℱadhLℋ.In view of Remark2, limLSℱ=∧U∈UℱlimLU, under the ultrafilter principle, in particular under the Axiom of Choice.Lemma 3.
LetL,limL be a convergence lattice. Under the Axiom of Choice, L,limL is Hausdorff if and only if L,limLS is Hausdorff.Proof.
SincelimLℱ≤limLSℱ, limLℱ is either ⊥ or an atom if this is the case for limLSℱ. Conversely, assume L,limLS is not Hausdorff, that is, there is ℱ∈FL and ℓ∈L with ⊥<ℓ<limLSℱ. Under the Axiom of Choice, this means that limLU>ℓ>⊥ for every U∈Uℱ and thus L,limL is not Hausdorff.Remark 6.
Note that the correspondingConv statement that a convergence ξ is Hausdorff if and only if its pseudotopological modification Sξ is, can be proved without invoking the ultrafilter principle, but we were not able to obtain the general point-free analog in a choice-free manner.
Another classical convergence result ([1], Corollary IX. 2.8), to the effect that a continuous bijection from a compact pseudotopology to a Hausdorff pseudotopology is a homeomorphism, finds a natural point-free generalization.Theorem 6.
Assume the Axiom of Choice. Letφ:L,limL⟶L′,limL′ be a bijective CConv morphism, L,limL be Hausdorff, and L′,limL′ be compact. Then, φ:L,limLS⟶L′,limL′S is an isomorphism.Keep in mind that ifφ is a bijective morphism of C, then U is a maximal filter if and only if φU is maximal, and φ preserves all infima and suprema.Proof.
We shall prove continuity ofφ−1, that is,(36)limLSℱ≤φ−1limL′Sφℱ,for all ℱ∈FL.
IfU∈UL then φU∈UL′ and thus limL′φU≠⊥ by compactness of L′. Moreover, limL′φU≤φlimLφ−1φU by continuity of φ applied to the filter φU, which is equivalent to φ−1limL′φU≤limLU because φ is bijective. Hence, ⊥≠φ−1limL′φU≤limLU. Because L,limL is Hausdorff, limLU is minimal in L\⊥ and therefore, φ−1limL′φU=limLU.
Now takeℱ∈FL. Applying the fact that φ is a C-isomorphism and thus φ−1 preserves all infima while Uφℱ=φU:U∈Uℱ, we obtain(37)limLSℱ=∧U∈UℱlimLU=∧U∈Uℱφ−1limL′φU=φ−1∧U∈UℱlimL′φU=φ−1∧W∈UφℱlimL′W=φ−1limL′Sφℱ.


### 3.3. Compact Morphisms

A morphismφ:L⟶L′ of CConv is compact if(38)φadhLφ−1ℱ≠⊥L′,for every ℱ∈FL′. This terminology comes from the following.Proposition 4.
A standard extremal epimorphismφ:L⟶L′ of CConv is compact if and only if its defining element ℓφ=∧φ−1⊤L′ is a compact element.Proof.
In view of Proposition 2, we may assumeL′=↓ℓφ and φm=m∧ℓφ. Hence, φ is compact if and only if ℓφ∧adhL↑ℱ≠⊥ for every ℱ∈F↓ℓφ, where ↑ℱ is the filter ℱ generates on L. Of course, it is equivalent to ask for ℓφ∧adhLG≠⊥ for every G∈FL with ℓφ∈G, for ℱ=g∧ℓφ:g∈G∈F↓ℓφ with ↑ℱ=G.
The following observation can be seen as an alternative abstract version (compare Theorem 4) of the fact that the continuous image of a compact space is compact (remember thatf:M⟶L is an abstraction of a map ℙg:ℙY⟶ℙX induced by a map g:X⟶Y).Proposition 5.
Iff:M⟶L is a CConv-morphism and φ:L⟶L′ is compact, then φ∘f:M⟶L′ is compact.Proof.
Letℱ∈FL′. We need to show that(39)φ∘fadhMf−1φ−1ℱ≠⊥L′,In view of Lemma 1, adhLφ−1ℱ≤fadhMf−1φ−1ℱ. Hence, by compactness of φ,(40)φ∘fadhMf−1φ−1ℱ≥φadhLφ−1ℱ>⊥L′.
In particular, in the case of a standard extremal epimorphismφ, we can identify L′ with ↓ℓ for a compact element ℓ∈L and φ with φm=m∧ℓ. That φ∘f is compact means that(41)fadhMf−1ℱ∧ℓ>⊥L,for every ℱ∈FL with ℓ∈ℱ. Now if f−1ℓ⊂G∈FL, then ℓ∈ℱ=fG, so that, in view of G⊂f−1fG,(42)fadhMG∧ℓ≥fadhMf−1fG∧ℓ>⊥L,and thus adhMG∧m>⊥M for every m∈f−1ℓ, that is, adhMG#f−1ℓ. In other words, f−1ℓ is a compact filter.
There are various remaining problems to consider related to compactness in the context of point-free convergence, most notably an analog of the Tychonoff theorem (which first requires an adequate notion mimicking products) and an analog of the Čech–Stone compactification. We hope to address these issues in a future work.


## 3.1. Compactness for Filters

Definition 4.
LetL,limL be a convergence lattice and D⊂FL. A filter ℱ∈FL is D-compact atA⊂L if(17)∀D∈D,D#ℱ⟹adhD∈A#.When D=FL, we omit the prefix D. When A=ℱ we omit “at ℱ.” Hence, ℱ is compact if(18)∀ℋ∈FL,ℋ#ℱ⟹adhℋ∈ℱ#.In the case where A=⊤, we omit “at A” and add the suffix “oid.” So ℱ is D-compactoid means that(19)∀ℋ∈D,ℋ#ℱ⟹adhℋ>⊥.
A related notion is that of nearD-compactness as introduced (in the case L=ℙX) in [5]. A filter ℱ on L is nearlyD-compact atA⊂L if(20)∀D∈D,D⊃ℱ⟹adhD∈A#.
Note that with these definitions, an elementℓ∈L is compact if and only if its principal filter ↑ℓ is a compact filter. Hence, relativizations of compactness with respect to a class of filters D (yielding notions of countable compactness, Lindelöfness, etc.) or with respect to a subset A of L can be applied to a single element, identifying it with its principal filter. As a result, we say that a convergence lattice L is D-compact if ⊤ is D-compactoid, that is, adhD>⊥ for every proper filter D∈D.Remark 3.
Note that whenD=F is the class of all filters, then D-compactness and near D-compactness are equivalent. Indeed, if ℱ is nearly compact (at A) and ℋ#ℱ there is a filter G finer than ℱ and ℋ. By near compactness, adhG∈A#. Since G⊃ℋ, adhG≤adhℋ and thus adhℋ∈A#.
It is clear that the image under theℙ functor of a compact convergence space is CConv-compact (that is, ℱ=⊤ is compact).Definition 5.
LetL,limL be a convergence lattice. A filter ℱ∈FL is closed if(21)adhℱ≤∧ℱ.Remark 4.
Note that this is a generalization to filters of the notion ofclosed element of a convergence lattice L,lim as defined in [4, 13] (which is different from the notion of closed element introduced in [2], where the present notion is called quasi-closed), where ℓ∈L is closed if(22)ℓ∈ℱ⟹limℱ≤ℓ,for every ℱ∈FL. Indeed, ℓ∈L is a closed element if and only if its principal filter ↑ℓ is closed in the sense of Definition 5.
Hence, a subspaceA,ξ|A of a convergence space X,ξ is closed if and only if A is a closed element of ℙX,limξ if and only if the principal filter of A in the convergence lattice ℙX,limξ is closed in the sense of the previous definition.Theorem 3.
LetL,limL be a convergence lattice and D⊂FL, let G⊃ℱ be filters on L, where ℱ is D-compactoid and G is closed. Then, G is nearly D-compact.Proof.
LetD∈D with D⊃G. Note that adhG≥adhD. Then, D#ℱ because G⊃ℱ, and ℱ is D-compactoid, hence,(23)adhG≥adhD>⊥.
AsG is closed, ⊥<adhD≤∧G so that(24)adhD∧g=adhD>⊥,for every g∈G and the conclusion follows.
In the case whereD=FL, we obtain, in view of Remark 3:Corollary 1.
Ifℱ is compactoid, G⊃ℱ, and G is closed then G is compact.In particular, in the caseℱ=⊤ we have the following.Corollary 2.
IfL,limL is a compact convergence lattice then every closed filter on L is compact.In particular, applying this fact to principal filters, the fact that closed subspaces of a compact convergence space are closed extends to the point-free setting: every closed element of a compact convergence lattice is compact.The point-free version of the classical fact that a continuous image of a compact set is compact will not extend straightforwardly to all morphisms in an arbitrary category of convergence lattices, but we can give a version in convergence frames.To this end, note that a class of filtersD consists of a set DL⊂FL of filters on each lattice L.Definition 6.
We say that a classD of filters is admissible if given a lattice morphism f:L⟶L′, fD∈DL′ whenever D∈DL and f−1ℋ∈DL whenever ℋ∈DL′.
Note that, in particular, ifD is admissible and L′⊂L, then every G∈DL′ generates a filter of DL.
Recall that frames are pseudocomplemented; that is, everyℓ∈L has a pseudocomplementℓ∗≔∨m∈L:m∧ℓ=⊥, which satisfies ℓ∧ℓ∗=⊥. Note that if ℓ has a pseudocomplement, then ℓ∈ℱ# if and only if ℓ∗∉ℱ. In general, a lattice or frame morphism does not need to preserve pseudocomplements, though φℓ∗≤φℓ∗ whenever ℓ and φℓ have pseudcomplements. On the other hand, a morphism of Heyting algebras preserves pseudocomplements.Lemma 2.
LetL and L′ be lattices and φ:L⟶L′ be a lattice morphism. Let ℱ∈FL′ and ℓ∈L. Then,(25)φℓ∈ℱ#⟹ℓ∈φ−1ℱ#.Ifℓis pseudocomplemented andφpreserves pseudocomplements, then the converse is true.Proof.
Ifφℓ∧m≠⊥ then ℓ∧a≠⊥ for every a∈φ−1m, for φℓ∧a=φℓ∧φa=φℓ∧m≠⊥ and φ⊥L=⊥L′. In particular, if φℓ∈ℱ# then ℓ∈φ−1ℱ#.
Conversely, ifℓ∈φ−1ℱ# then ℓ∗∉φ−1ℱ, that is, φℓ∗∉ℱ. If φ respects pseudocomplements φℓ∗=φℓ∗∉ℱ equivalently, φℓ∈ℱ#.Theorem 4.
Letg:L,limL⟶H,limH be a FrmConv morphism that preserves pseudocomplements. If ℱ∈FH is compact, then g−1ℱ is compact.Note that in the classical case whereL=ℙY, H=ℙX, and g=f−1, where f:X⟶Y, then L and H are Boolean algebras, hence complemented, and g respects complements, hence pseudocomplements.Proof.
Letℓ∈L such that gℓ∈ℱ. If ℋ#g−1ℱ, then gℋ#ℱ by Lemma 2. Thus, adhgℋ#ℱ because ℱ is compact. In particular, adhgℋ∧gℓ>⊥, that is,(26)∨W#gℋlimW∧gℓ≠⊥.
SinceL is a frame, this implies that there is W#gℋ with limW∧gℓ≠⊥.
Now,limW≤glimg−1W by (ptfree continuity). Thus,(27)⊥≠limW∧gℓ≤glimg−1W∧ℓ,so that limg−1W∧ℓ≠⊥, because g is a lattice morphism, hence sends ⊥ to ⊥.
Recall that a convergence space isHausdorff if limℱ consists of at most one point for every filter ℱ on the space. The following definition gives an abstraction to the point-free setting.Definition 7.
We say a convergence latticeL,limL is Hausdorff if for every ℱ∈FL either limℱ=⊥ or limℱ is an atom of L; that is, limℱ is minimal in L\⊥.
It is straightforward that the image underℙ of any Hausdorff convergence space is a Hausdorff convergence lattice.Theorem 5.
LetL,limL be a Hausdorff convergence frame and let ℱ be a compact filter on L. Then, ℱ is closed.Proof.
We have to prove that(28)∨ℱ⊂GlimG=adhℱ≤∧ℱ.
It will be enough to prove that(29)limG≤f,for any G⊃ℱ and any f∈ℱ. For any such G and f, adhG#ℱ by compactness of ℱ. Hence,(30)∨ℋ⊃Glimℋ∧f>⊥,equivalently,(31)∨ℋ⊃Glimℋ∧f>⊥,because L is a frame. Hence,(32)∀f∈ℱ∃ℋf⊃Glimℋf∧f>⊥.
BecauseL is Hausdorff, it follows that(33)limG≤limℋf=limℋf∧f≤f.Remark 5.
The same result can be obtained without using the assumption thatL be a frame, using maximal filters as in Remark 2, hence using the Axiom of Choice.


## 3.2. Pseudotopological Convergence Lattices and Minimality of Compact Hausdorff Structures

We call a convergence latticepseudotopological, if for every ℱ∈FL,(34)limℱ≥∧ℋ#ℱadhℋ.Note that the reverse inequality in (34) is always true (of course, (34) implicitly assumes that the infimum involved exists. Hence, pseudotopological convergence lattices are most naturally considered in the context of complete lattices).Given a convergence latticeL,limL, we define its pseudotopological modification limLS:FL⟶L by(35)limLSℱ≔∧ℋ#ℱadhLℋ.In view of Remark2, limLSℱ=∧U∈UℱlimLU, under the ultrafilter principle, in particular under the Axiom of Choice.Lemma 3.
LetL,limL be a convergence lattice. Under the Axiom of Choice, L,limL is Hausdorff if and only if L,limLS is Hausdorff.Proof.
SincelimLℱ≤limLSℱ, limLℱ is either ⊥ or an atom if this is the case for limLSℱ. Conversely, assume L,limLS is not Hausdorff, that is, there is ℱ∈FL and ℓ∈L with ⊥<ℓ<limLSℱ. Under the Axiom of Choice, this means that limLU>ℓ>⊥ for every U∈Uℱ and thus L,limL is not Hausdorff.Remark 6.
Note that the correspondingConv statement that a convergence ξ is Hausdorff if and only if its pseudotopological modification Sξ is, can be proved without invoking the ultrafilter principle, but we were not able to obtain the general point-free analog in a choice-free manner.
Another classical convergence result ([1], Corollary IX. 2.8), to the effect that a continuous bijection from a compact pseudotopology to a Hausdorff pseudotopology is a homeomorphism, finds a natural point-free generalization.Theorem 6.
Assume the Axiom of Choice. Letφ:L,limL⟶L′,limL′ be a bijective CConv morphism, L,limL be Hausdorff, and L′,limL′ be compact. Then, φ:L,limLS⟶L′,limL′S is an isomorphism.Keep in mind that ifφ is a bijective morphism of C, then U is a maximal filter if and only if φU is maximal, and φ preserves all infima and suprema.Proof.
We shall prove continuity ofφ−1, that is,(36)limLSℱ≤φ−1limL′Sφℱ,for all ℱ∈FL.
IfU∈UL then φU∈UL′ and thus limL′φU≠⊥ by compactness of L′. Moreover, limL′φU≤φlimLφ−1φU by continuity of φ applied to the filter φU, which is equivalent to φ−1limL′φU≤limLU because φ is bijective. Hence, ⊥≠φ−1limL′φU≤limLU. Because L,limL is Hausdorff, limLU is minimal in L\⊥ and therefore, φ−1limL′φU=limLU.
Now takeℱ∈FL. Applying the fact that φ is a C-isomorphism and thus φ−1 preserves all infima while Uφℱ=φU:U∈Uℱ, we obtain(37)limLSℱ=∧U∈UℱlimLU=∧U∈Uℱφ−1limL′φU=φ−1∧U∈UℱlimL′φU=φ−1∧W∈UφℱlimL′W=φ−1limL′Sφℱ.


## 3.3. Compact Morphisms

A morphismφ:L⟶L′ of CConv is compact if(38)φadhLφ−1ℱ≠⊥L′,for every ℱ∈FL′. This terminology comes from the following.Proposition 4.
A standard extremal epimorphismφ:L⟶L′ of CConv is compact if and only if its defining element ℓφ=∧φ−1⊤L′ is a compact element.Proof.
In view of Proposition 2, we may assumeL′=↓ℓφ and φm=m∧ℓφ. Hence, φ is compact if and only if ℓφ∧adhL↑ℱ≠⊥ for every ℱ∈F↓ℓφ, where ↑ℱ is the filter ℱ generates on L. Of course, it is equivalent to ask for ℓφ∧adhLG≠⊥ for every G∈FL with ℓφ∈G, for ℱ=g∧ℓφ:g∈G∈F↓ℓφ with ↑ℱ=G.
The following observation can be seen as an alternative abstract version (compare Theorem 4) of the fact that the continuous image of a compact space is compact (remember thatf:M⟶L is an abstraction of a map ℙg:ℙY⟶ℙX induced by a map g:X⟶Y).Proposition 5.
Iff:M⟶L is a CConv-morphism and φ:L⟶L′ is compact, then φ∘f:M⟶L′ is compact.Proof.
Letℱ∈FL′. We need to show that(39)φ∘fadhMf−1φ−1ℱ≠⊥L′,In view of Lemma 1, adhLφ−1ℱ≤fadhMf−1φ−1ℱ. Hence, by compactness of φ,(40)φ∘fadhMf−1φ−1ℱ≥φadhLφ−1ℱ>⊥L′.
In particular, in the case of a standard extremal epimorphismφ, we can identify L′ with ↓ℓ for a compact element ℓ∈L and φ with φm=m∧ℓ. That φ∘f is compact means that(41)fadhMf−1ℱ∧ℓ>⊥L,for every ℱ∈FL with ℓ∈ℱ. Now if f−1ℓ⊂G∈FL, then ℓ∈ℱ=fG, so that, in view of G⊂f−1fG,(42)fadhMG∧ℓ≥fadhMf−1fG∧ℓ>⊥L,and thus adhMG∧m>⊥M for every m∈f−1ℓ, that is, adhMG#f−1ℓ. In other words, f−1ℓ is a compact filter.
There are various remaining problems to consider related to compactness in the context of point-free convergence, most notably an analog of the Tychonoff theorem (which first requires an adequate notion mimicking products) and an analog of the Čech–Stone compactification. We hope to address these issues in a future work.



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