Pointfree Setting Research Articles

Subobjects and Compactness in Point-Free Convergence

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.

Adjoint maps between implicative semilattices and continuity of localic maps

We study residuated homomorphisms (r-morphisms) and their adjoints, the so-called localizations (or l-morphisms), between implicative semilattices, because these objects may be characterized as semilattices whose unary meet operations have adjoints. Since left resp. right adjoint maps are the residuated resp. residual maps (having the property that preimages of principal downsets resp. upsets are again such), one may not only regard the l-morphisms as abstract continuous maps in a pointfree framework (as familiar in the complete case), but also characterize them by concrete closure-theoretical continuity properties. These concepts apply to locales (frames, complete Heyting lattices) and provide generalizations of continuous and open maps between spaces to an algebraic (not necessarily complete) pointfree setting.

Sequences and related closure and compactness properties in frames

The aim of this note is to introduce concepts of sequence and sequence convergence in the point-free setting. These are then used to define sequential closedness and associated compactness properties in frames. The interplay of these closure and compactness notions are investigated. Relating back to the pointed setting provides additional insight into an asymmetric Kuratowski-Mrówka type theorem for countable compactness in general topology.

On an aspect of scatteredness in the point-free setting

It is well known that a locale is subfit iff each of its open sublocales is a join of closed ones, and fit iff each of its closed sublocales is a meet of open ones. This formulation, however, exaggerates the parallelism between the behavior of fitness and subfitness. For it can be shown that a locale is fit iff each of its sublocales is a meet of closed ones, but it is not the case that a locale is subfit iff each of its sublocales is a join of closed ones. Thus we are led to take up the very natural question of which locales have the feature that every sublocale is a join of closed sublocales. In this note we show that these are precisely the subfit locales which are scattered in the point-free sense of [13], and we add a variation for spatial frames.

Normal semicontinuity and the Dedekind completion of pointfree function rings

This paper supplements an earlier one by the authors which constructed the Dedekind completion of the ring of continuous real functions on an arbitrary frame L in terms of partial continuous real functions on L. In the present paper, we provide three alternative views of it, in terms of (i) normal semicontinuous real functions on L, (ii) the Booleanization of L (in the case of bounded real functions) and the Gleason cover of L (in the general case), and (iii) Hausdorff continuous partial real functions on L. The first is the normal completion and extends Dilworth’s classical construction to the pointfree setting. The second shows that in the bounded case, the Dedekind completion is isomorphic to the lattice of bounded continuous real functions on the Booleanization of L, and that in the non-bounded case, it is isomorphic to the lattice of continuous real functions on the Gleason cover of L. Finally, the third is the pointfree version of Anguelov’s approach in terms of interval-valued functions. Two new classes of frames, cb-frames and weak cb-frames, emerge naturally in the first two representations. We show that they are conservative generalizations of their classical counterparts.

Tightness relative to some (co)reflections in topology

We address what might be termed the reverse reflection problem: given a monoreflection from a category A onto a subcategory B, when is a given object B ∈ B the reflection of a proper subobject? We start with a well known specific instance of this problem, namely the fact that a compact metric space is never the Čech-Stone compactification of a proper subspace. We show that this holds also in the pointfree setting, i.e., that a compact metrizable locale is never the Čech-Stone compactification of a proper sublocale. This is a stronger result than the classical one, but not because of an increase in scope; after all, assuming weak choice prin­ciples, every compact regular locale is the topology of a compact Hausdorff space. The increased strength derives from the conclusion, for in general a space has many more sublocales than subspaces. We then extend the analysis from metric locales to the broader class of perfectly normal locales, i.e., those whose frame of open sets consists entirely of cozero elements. We include a second proof of these results which is purely algebraic in character.At the opposite extreme from these results, we show that an extremally disconnected locale is a compactification of each of its dense sublocales. Finally, we analyze the same phenomena, also in the pointfree setting, for the 0-dimensional compact reflec­tion and for the Lindelöf reflection.

Pointfree pointwise suprema in unital archimedean ℓ-groups

We generalize the concept of the pointwise supremum of real-valued functions to the pointfree setting. The concept itself admits a direct and intuitive formulation which makes no mention of points. But our aim here is to investigate pointwise suprema of subsets of RL, the family of continuous real valued functions on a locale, or pointfree space.Thus our setting is the category W of archimedean lattice-ordered groups (ℓ-groups) with designated weak order unit, with morphisms which preserve the group and lattice operations and take units to units. This is an appropriate context for this investigation because every W-object can be canonically represented as a subobject of some RL.We show that the suprema which are pointwise in the Madden representation can be characterized purely algebraically. They are precisely the suprema which are context-free, in the sense of being preserved by every W homomorphism out of G. We show that closure under such suprema characterizes the W-kernels among the convex ℓ-subgroups. Finally, we prove that all existing joins in a W-object G are pointwise iff its Madden frame L is boolean, and that all existing countable joins in G are pointwise if L is a P-frame, but not conversely.This leads up to the appropriate analog of the Nakano–Stone Theorem: a (completely regular) locale L has the feature that RL is conditionally pointwise complete (σ-complete), i.e., every bounded (countable) family from RL has a pointwise supremum in RL, iff L is boolean (a P-locale).We adopt a maximally broad definition of unconditional pointwise completeness (σ-completeness): a divisible W-object G is pointwise complete (σ-complete) if it contains a pointwise supremum for every subset which has a supremum in any extension. We show that the pointwise complete (σ-complete) W-objects are those of the form RL for L a boolean locale (P-locale). Finally, we show that a W-object G is pointwise σ-complete iff it is epicomplete.

Insertion and extension results for pointfree complete regularity

There are insertion-type characterizations in pointfree topology that extend well known insertion theorems in point-set topology for all relevant higher separation axioms with one notable exception: complete regularity. In this paper we fill this gap. The situation reveals to be an interesting and peculiar one: contrarily to what happens with all the other higher separation axioms, the extension to the pointfree setting of the classical insertion result for completely regular spaces characterizes a formally weaker class of frames introduced in this paper (called \emph{completely c-regular frames}). The fact that any compact sublocale (quotient) of a completely regular frame is a $C$-sublocale ($C$-quotient) is obtained as a corollary.

Realcompact Alexandroff frames

This paper complements an earlier one by the authors which showed, among other things, that every completely regular σ-frame is complete in the natural uniformity associated with its continuous real-valued functions — very much a characteristic property associated with realcompactness. It is natural then, when exploiting the pointfree setting for realcompactness, to rather work with Alexandroff frames, which we show here are sufficiently close to completely regular frames to maintain the distinctions between the different notions of realcompactness found there, but nevertheless still retain desirable properties analogous to those that their classical archetypes the Alexandroff spaces have in contrast to their topological counterparts.

Asymmetric filter convergence and completeness

Completeness for metric spaces is traditionally presented in terms of convergence of Cauchy sequences, and for uniform spaces in terms of Cauchy filters. Somewhat more abstractly, a uniform space is complete if and only if it is closed in every uniform space in which it is embedded, and so isomorphic to any space in which it is densely embedded. This is the approach to completeness used in the point-free setting, that is, for uniform and nearness frames: a nearness frame is said to be complete if every strict surjection onto it is an isomorphism. Quasi-uniformities and quasi-nearnesses on biframes provide appropriate structures with which to investigate uniform and nearness ideas in the asymmetric context. In [9] a notion of completeness (called “quasi-completeness”) was presented for quasi-nearness biframes in terms of suitable strict surjections being isomorphisms, and a quasi-completion was constructed for any quasi-nearness biframe. In this paper we show that quasi-completeness can indeed be viewed in terms of the convergence of certain filters, namely, the regular Cauchy bifilters. We use the notion of a T -valued bifilter, which generalizes the characteristic function of a filter. An important tool is an appropriate composition for such bifilters. We show that the right adjoint of the quasi-completion is the universal regular Cauchy bifilter and use it to prove this characterization of quasi-completeness. We also construct the so-called Cauchy filter quotient for a biframe using a quotient of the downset biframe that involves only the Cauchy, and not the regularity, condition. Like the quasi-completion, this provides a universal Cauchy bifilter; unlike the quasi-completion, this construction is functorial.

Rings of continuous functions on σ-frames

This paper deals with the ℓ-rings R S of all real-valued continuous functions on a completely regular σ-frame. It shows that, in marked contrast with the situation for frames, any ℓ-ring homomorphism R S → R T results from a σ-frame homomorphism S → T . Further, it proves the analogue of this for integer-valued continuous functions and 0-dimensional σ-frames. In all, this demonstrates that the important classical difference between Alexandroff spaces and Tychonoff spaces with respect to the real-valued continuous functions carries over fully to the pointfree setting – indeed, it adds the integer-valued case which seems to be new in this context.

Lower and upper regularizations of frame semicontinuous real functions

As discovered recently, Li and Wang’s 1997 treatment of semicontinuity for frames does not faithfully reflect the classical concept. In this paper we continue our study of semicontinuity in the pointfree setting. We define the pointfree concepts of lower and upper regularizations of frame semicontinuous real functions. We present characterizations of extremally disconnected frames in terms of these regularizations that allow us to reprove, in particular, the insertion and extension type characterizations of extremally disconnected frames due to Y.-M. Li and Z.-H. Li [Algebra Universalis 44 (2000), 271–281] in the right semicontinuity context. It turns out that the proof of the insertion theorem becomes very easy after having established a number of basic results regarding the regularizations. Notably, our extension theorem is a much strengthened version of Li and Li's result and it is proved without making use of the insertion theorem.

The Stone–Čech Compactification and the Cozero Lattice in Pointfree Topology

The existence of the pointfree Stone–Cech compactification, here understood as the compact, completely regular coreflection of a frame, has been established long ago in several ways modelled after certain descriptions of the classical Stone–Cech compactification (Johnstone, Stone Spaces, Cambridge Stud. Adv. Math. 3. Cambridge University Press, Cambridge, 1982; Banaschewski and Mulvey, Houston J. Math. 6:301–312, 1980, J. Pure Appl. Algebra 33:107–122, 1984), but – somewhat surprisingly – one of the most familiar representations of the latter, the one popularized by Gillman and Jerison’s influential book as the space of maximal z-filters, has only recently been recognized also to carry over to the pointfree setting, as is implicit in Wei (Appl. Categ. Structures 12:197–202, 2004). The purpose of this note is to present an alternative account of this fact which places the problem into the wider context of the saturation quotients of compact frames and what are introduced here as completely normal lattices. Specifically, we first show that the saturation quotient S(Open image in new window A) of the frame Open image in new window A of ideals of a completely normal lattice A is compact completely regular (Proposition 1) which is then brought to bear on the question at hand by establishing that the cozero lattice CozL of a frame L is completely normal and finally leads to the result that SOpen image in new window(CozL) is the Stone–Cech compactification of L (Proposition 2). Given that familiar results concerning saturation quotients exhibit the spectrum of the latter frame as the space of maximal z-filters of X for the cozero set lattice of a topological space X, this is then exactly the pointfree form of the desired kind. With regard to foundations, it should be emphasized that, in notable contrast with most other work involving CozL, the present arguments are constructively valid in the sense of topos theory.

Well-Embedding and G δ -Density in a Pointfree Setting

Translating the notion of Gδ-dense and well-embedded subspaces into a pointfree setting gives some elegant results (with remarkably simple proofs), most notably how the two concepts are related – a fact which, no doubt, is well-known to classical topologists, but is easily overlooked.

Coz-onto Frame Maps and Some Applications

A frame homomorphism is coz-onto if it maps the cozero part of its domain surjectively onto that of its codomain. This captures the notion of a z-embedded subspace of a topological space in a point-free setting. We give three different types of characterizations of coz-onto homomorphisms. The first is in terms of elements, the second in terms of quotients, and the last in terms of ideals. As an application of properties of coz-onto homomorphisms developed herein, we present some characterizations of \(F\)- and \(F^{\prime }\)-frames.

Recursive Quantum Functions, Avoidable Points, & Shadow Points in Recursive Analysis

In this paper we summarize our approach and findings in a point-free setting for recursive topology and recursive analysis. Recursive analysis has received intensive attention recently and our approach, while differing from those of other schools, does have connections to them. Here, we also present some of our findings on quantum and total recursive functions on the reals, introduce our classification of nonrecursive points, and remark on the connections between our work and the works of others.

A Note on Total Boundedness

In the pointfree setting of frames, some new characterizations for the old notion of total boundedness are presented. Applications are made to normal and compact regular frames.