Jorge Picado

Frames and Locales

Preface.- Introduction.- I. Spaces and lattices of open sets.- II. Frames and locales. Spectra.- III. Sublocales.- IV. Structure of localic morphisms. The categories Loc and Frm.- V. Separation axioms.- VI. More on sublocales.-VII. Compactness and local compactness.- VIII. (Symmetric) uniformity and nearness.- IX. Paracompactness.- X. More about completion.- XI. Metric frames.- XII. Entourages, non-symmetric uniformity.- XIII. Connectedness.- XIV. The frame of reals and real functions.- XV. Localic groups.- Appendix I: Posets.- Appendix II: Categories.- Bibliography.- Index of Notation.- Index.

Structured Frames by Weil Entourages

In the framework of pointfree topology, we discuss the role of Weil entourages in the study of structures such as uniformities, quasi-uniformities, nearnesses, quasi-nearnesses, proximities and infinitesimal relations.

Functorial Quasi-Uniformities on Frames

We present a unified study of functorial quasi-uniformities on frames by means of Weil entourages and frame congruences. In particular, we use the pointfree version of the Fletcher construction, introduced by the authors in a previous paper, to describe all functorial transitive quasi-uniformities.

Uniform-type structures on lattice-valued spaces and frames

By introducing lattice-valued covers of a set, we present a general framework for uniform structures on very general L-valued spaces (for L an integral commutative quantale). By showing, via an intermediate L-valued structure of uniformity, how filters of covers may describe the uniform operators of Hutton, we prove that, when restricted to Girard quantales, this general framework captures a significant class of Huttons uniform spaces. The categories of L-valued uniform spaces and L-valued uniform frames here introduced provide (in the case L is a complete chain) the missing vertices in the commutative cube formed by the classical categories of topological and uniform spaces and their corresponding pointfree counterparts (forming the base of the cube) and the corresponding L-valued categories (forming the top of the cube).

More on Subfitness and Fitness

The concepts of fitness and subfitness (as defined in Isbell, Trans. Amer. Math. Soc. 327, 353–371, 1991) are useful separation properties in point-free topology. The categorical behaviour of subfitness is bad and fitness is the closest modification that behaves well. The separation power of the two, however, differs very substantially and subfitness is transparent and turns out to be useful in its own right. Sort of supplementing the article (Simmons, Appl. Categ. Struct. 14, 1–34, 2006) we present several facts on these concepts and their relation. First the “supportive” role subfitness plays when added to other properties is emphasized. In particular we prove that the numerous Dowker-Strauss type Hausdorff axioms become one for subfit frames. The aspects of fitness as a hereditary subfitness are analyzed, and a simple proof of coreflectivity of fitness is presented. Further, another property, prefitness, is shown to also produce fitness by heredity, in this case in a way usable for classical spaces, which results in a transparent characteristics of fit spaces. Finally, the properties are proved to be independent.

Entourages, Covers and Localic Groups

Due to the nature of product in the category of locales, the entourage uniformities in the point-free context only mimic the classical Weil approach while the cover (Tukey type) ones can be viewed as an immediate extension. Nevertheless the resulting categories are concretely isomorphic. We present a transparent construction of this isomorphism, and apply it to the natural uniformities of localic groups. In particular we show that localic group homomorphisms are uniform, thus providing natural forgetful functors from the category of localic groups into any of the two categories of uniform locales.

On Strong Inclusions and Asymmetric Proximities in Frames

The strong inclusion, a specific type of subrelation of the order of a lattice with pseudocomplements, has been used in the concrete case of the lattice of open sets in topology for an expedient definition of proximity, and allowed for a natural pointfree extension of this concept. A modification of a strong inclusion for biframes then provided a pointfree model also for the non-symmetric variant. In this paper we show that a strong inclusion can be non-symmetrically modified to work directly on frames, without prior assumption of a biframe structure. The category of quasi-proximal frames thus obtained is shown to be concretely isomorphic with the biframe based one, and shown to be related to that of quasi-uniform frames in a full analogy with the symmetric case.

On two extension of dicksons torsion theory

We establish relations between two extensions of Dicksons torsion theory, one introduced by Barr and the other by Cassidy, Hubert and Kelly. These two notions are studied in a general category and we show that categories with zero object are, in some sense, the ones where the second notion is relevant. In these categories, under suitable completeness conditions, we characterize torsion theories in terms of reflections and connections. A characterization of the torsion-free subcategories which generalizes the one of Dickson is also presented.

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 pointfree sense of (13), and we add a variation for spatial frames.

Notes on Exact Meets and Joins

An exact meet in a lattice is a special type of infimum characterized by, inter alia, distributing over finite joins. In frames, the requirement that a meet is preserved by all frame homomorphisms makes for a slightly stronger property. In this paper these concepts are studied systematically, starting with general lattices and proceeding through general frames to spatial ones, and finally to an important phenomenon in Scott topologies.