- S. Maurer
Sometimes I don't understand how people came across the concept of "fun"; it was probably only abstracted as an opposite to sadness.
- F. Kafka
In the thirties, Stone presented in the famous papers [The theory of representation for Boolean algebras, Trans. Amer. Math. Soc. 40 (1936) 37-111] and [Applications of the theory of Boolean rings to general topology, Trans. Amer. Math. Soc. 41 (1937) 375-481] two revolutionary ideas. Firstly, he concluded that ideals are very important in lattice theory, by showing that Boolean algebras are actually special instances of rings: the concept of Boolean algebra is equivalent to a certain type of ring --- nowadays called Boolean ring. Afterwards, following his maxim "one must always topologize" [The representation of Boolean algebras, Bull. Amer. Math. Soc. 44 (1938) 807-816], he linked topology to lattice theory by establishing a representation theorem for Boolean algebras:
Every Boolean algebra is isomorphic to the Boolean algebra of open-closed sets of a totally disconnected compact Hausdorff space (or, in other words, compact zero-dimensional Hausdorff space).
This theorem has had a great influence in many areas of modern mathematics (the reader may see a detailed description of it in the book [Stone spaces, Cambridge University Press, 1982] by Johnstone), namely in the study of topological concepts from a lattice theoretical point of view, initiated with Wallman in 1938 and followed with McKinsey and Tarski (1944), Nöbeling (1954), Lesieur (1954), Ehresmann (1957), Dowker and Papert (1966), Banaschewski (1969), Isbell (1972), Simmons (1978), Johnstone (1981), Pultr (1984), among others.
Ehresmann and Bénabou were the first (in 1957) to look at complete lattices with an appropriate distributive law --- finite meets distribute over arbitrary joins --- as "generalized" topological spaces. They called these lattices local lattices, meanwhile, named frames after Dowker and Papert.
Frame theory is lattice theory applied to topology. This approach to topology takes the lattices of open sets as the basic notion --- it is a "pointfree topology". There, one investigates typical properties of lattices of open sets that can be expressed without reference to points.
Usually one thinks of frames as generalized spaces:
"The generalized spaces will
be called locales.
"Generalized" is imprecise, since arbitrary spaces are not
determined by their
lattices of open sets; but the "insertion" from spaces to locales is
full and faithful on Hausdorff spaces"
[Isbell, Atomless parts of spaces, Math. Scand. 31 (1972) 5-32].
Nevertheless, the frame homomorphisms --- which should preserve finite meets and arbitrary joins --- may only be interpreted as "generalized continuous maps" when considered in the dual category. Isbell, in his celebrated paper of 1972 "Atomless parts of spaces", was the first to stress this and to point out the need for a separate terminology for the dual category of frames, whose objects he named "locales", as cited above.
Johnstone in [Stone spaces, Cambridge University Press, 1982], [The point of pointless topology, Bull. Amer. Math. Soc. (N.S.) 8 (1983) 41-53] and [The Art of pointless thinking: a student's guide to the category of locales, in Category Theory at Work (Proc. Workshop Bremen 1990), Heldermann, 1991, pp. 85-107] gives us a detailed account of these historical developments and of the advantages of this new way of doing topology as opposed to the classical one. As Isbell states in his review of [Johnstone, The Art of pointless thinking: a student's guide to the category of locales, in Category Theory at Work (Proc. Workshop Bremen 1990), Heldermann, 1991, pp. 85-107] in "Zentralblatt für Mathematik",
"this paper is an argument that topology is better modeled in the category of locales than in topological spaces or another of their variants, with indication of how the millieu should be regarded and supporting illustrations".
Frame theory has the advantage that many results which require the Axiom of Choice (or some of its variants) in the topological setting may be proved constructively in the frame setting. Examples are the Tychonoff Theorem [Johnstone, Tychonoff's Theorem without the Axiom of Choice, Fund. Math. 113 (1981) 21-35], the construction of the Stone-Cech compactification [Banaschewski and Mulvey, Stone-Cech compactification of locales, I, Houston J. Math. 6 (1980) 301-312] or the construction of the Samuel compactification [Banaschewski and Pultr, Samuel compactification and completion of uniform frames, Math. Proc. Cambridge Philos. Soc. 108 (1990) 63-78]. By that reason, locales are the "right" spaces for topos theory [MacLane and Moerdijk, Sheaves in Geometry and Logic: A First Introduction to Topos Theory, Springer, 1992].
Sometimes the frame situation differs from the classical one. In general, when this happens, the frame situation is more convenient. For example, coproducts of paracompact frames are paracompact [Isbell, Atomless parts of spaces, Math. Scand. 31 (1972) 5-32] while products of paracompact spaces are not necessarily paracompact. Another example: coproducts of regular frames preserve the Lindelöf property [Dowker and Strauss, Sums in the category of frames, Houston J. Math. 3 (1977) 7-15], products of regular spaces do not.
One may also look to frames as the type of algebras behind the "logic of affirmative assertions". This is the approach of Vickers in "Topology via Logic" [Cambridge Tracts in Theoretical Computer Science 5, Cambridge University Press, 1989]:
"The traditional --- spatial --- motivation for general topology and its
axioms relies on abstracting first from Euclidean space to metric spaces, and
then abstracting out, for no obvious reason, certain properties of their
open sets. I believe that the localic view helps to clarify these
axioms, by interpreting them not as set theory (finite intersections and
arbitrary unions), but as logic (finite conjunctions and arbitrary
disjunctions: hence the title)".
... "I have tried to argue directly from these logical intuitions to the topological axioms, and to frames as the algebraic embodiment of them".
The study of structured frames started with Isbell [Atomless parts of spaces, Math. Scand. 31 (1972) 5-32], who considered the notion of frame uniformity in the form of a system of covers, later developed by Pultr in [Pointless uniformities I. Complete regularity, Comment. Math. Univ. Carolin. 25 (1984) 91-104] and [Pointless uniformities II. (Dia)metrizatio, Comment. Math. Univ. Carolin. 25 (1984) 105-120], who also defined metric diameters for frames, using them in analogy with pseudometrics in the spatial setting. Subsequently, Frith [Structured Frames, Doctoral Dissertation, University of Cape Town, 1987] studied uniform-like structures from a more categorical point of view, introducing in frame theory other topological structures such as quasi-uniformities and proximities. Most of these concepts are formulated in terms of covers, and indeed Frith even stated that
"families of covers constitute the only tool that works for frames"
[Structured Frames, Doctoral Dissertation, University of Cape Town, 1987].
Nevertheless, Fletcher and Hunsaker [Entourage uniformities for frames, Monatsh. Math. 112 (1991) 271-279] recently presented an equivalent notion of frame uniformity in terms of certain families of maps from the frame into itself.
This dissertation has its origin in the following suggestion of Professor Bernhard Banaschewski:
"We usually consider uniformities given by covers, as done by Tukey for spaces, but there should also be a theory (deliberately put aside by Isbell in "Atomless parts of spaces") of uniformities by entourages, in the style of Bourbaki".
So, the study of structured frames via entourages in the style of Weil is the subject of this thesis. We begin with uniform structures (Chapter I) and in Chapters III and IV we investigate the corresponding natural generalizations of quasi-uniform and nearness structures. In parallel, with the aim of completing for frames a picture analogous to the one for spaces, we characterize uniform structures in terms of "gauge structures", that is, families of metric diameters satisfying certain axioms.
The language of this thesis is almost entirely algebraic and we never employ the geometric approach made possible by the language of locales. This point of view is corroborated by Madden [k-frames, J. Pure Appl. Algebra 70 (1991) 107-127]:
"There are differing opinions about this, and I appreciate that there are some very good reasons for wanting to keep the geometry in view. On the other hand, the algebraic language seems to me, after much experimentation, to afford the simplest and most streamlined presentation of results. Also, I think readers will not have much difficulty finding the geometric interpretations themselves, if they want them. After all, this ultimately comes down to just "reversing all the arrows" ".
Throughout this dissertation we also adopt the categorical point of view. The language of category theory has proved to be an adequate tool in the approach to the type of conceptual problems we are interested in, namely in the selection of the best axiomatizations for some frame structures and in the study of the relationship between them. Furthermore, this insight allows us to understand and to put in perspective the real meaning of these approaches. As Herrlich and Porst state in the preface to "Category Theory at Work":
"Some mathematical concepts appear to be "unavoidable", e.g. that of natural numbers. For other concepts such a claim seems debatable, e.g., for the concepts of real numbers or of groups. Other concepts --- within certain limits --- seem to be quite arbitrary, their use being based more on historical accidents than on structural necessities. A good example is the concept of topological spaces: compare such "competing" concepts as metric spaces, convergence spaces, pseudotopological spaces, uniform spaces, nearnes3 spaces, frames respectively locales, etc. What are the structural "necessities" or at least "desirabilities"? Category theory provides a language to formulate such questions with the kind of precision needed to analyse advantages and disadvantages of various alternatives. In particular, category theory enables us to decide whether certain mathematical "disharmonies" are due to inherent structural features or rather to chance ocurrences, and in the latter case helps to "set things right" ".
This is the point of view adopted in this work as well as in the articles [Picado, Weil uniformities for frames, Comment. Math. Univ. Carolin. 36 (1995) 357-370], [Picado, Frame quasi-uniformities by entourages, to appear in Festschrift for G.C.L. Brümmer on his 60th birthday, University of Cape Town], and [Picado, Weil nearnesses for frames and spaces, Preprint 95-21, University of Coimbra, 1995] on which it relies. Another thread of our work is the search, in each setting, for an adjunction between structured versions of the "open" and "spectrum" functors. These functors acted as a categorical guide of the accuracy of the choosen axiomatizations.
We describe now this thesis in more detail:
Chapter 0 introduces well-known basic definitions and results needed in the body of the dissertation.
Chapter I, which is the core of this work, presents a theory of frame uniformities in the style of Weil (Section 4) which is equivalent to the ones of Isbell [Atomless parts of spaces, Math. Scand. 31 (1972) 5-32] and Fletcher and Hunsaker [Entourage uniformities for frames, Monatsh. Math. 112 (1991) 271-279], as is proved in Theorem 5.14. The chapter ends with an application of this theory to the study, in the setting of frames, of an important theorem of the theory of uniform spaces, due to Efremovic. Our approach via entourages reveals to be the right language to yield the result analogous to the one of Efremovic in the context of frames.
After Chapter I it would be natural to investigate the non-symmetric structures (as well as the nearness structures) that arise from our theory of uniformities. However the existence of another characterization of uniform spaces due to Bourbaki [Topologie Générale, Livre III, Chapitre 9, Hermann, 1948] and the notion of metric diameter introduced by Pultr [Pointless uniformities II. (Dia)metrization, Comment. Math. Univ. Carolin. 25 (1984) 105-120] lead us to investigate a way of describing frame uniformities in terms of those diameters. This is what it is done in the beginning of Chapter II. As an aplication of this characterization, inspired by the paper [The category of uniform spaces as a completion of the category of metric spaces, Comment. Math. Univ. Carolin. 33 (1992) 689-693] of Adámek and Reiterman, it is shown in Corollary 4.20 that the category of uniform frames is fully embeddable in a (final and universal) completion of the category of metric frames. Therefore, metric frames provide a categorical motivation for uniform frames.
In Chapters III and IV we come back to the main stream of our work. The third section of Chapter III contains the axiomatization of a theory of quasi-uniformities via Weil entourages. Theorem 4.15 shows that this is an equivalent theory to the existing one in the literature.
In the final chapter we proceed to another level of generality by studying nearness structures in frames using Weil entourages. In this case the corresponding spatial structures, which seem unnoticed so far, appear as a topic worthy of study. Although distinct from the classical nearness spaces of Herrlich [A concept of nearness, General Topology and Appl. 4 (1974) 191-212], this class of spaces forms a nice topological category (Proposition 5.1) which unifies several topological and uniform concepts (Propositions 5.4, 5.5, 5.6, 5.8 and Corollary 5.15). The notion of Weil entourage is therefore a basic topological concept by means of which various topological ideas may be expressed. In the last section we study proximal frames. Theorem 6.10 gives a new characterization of this type of frames in terms of Weil entourages. The infinitesimal relations of Efremovic [Infinitesimal spaces, Dokl. Akad. Nauk 76 (1951) 341-343] are also studied in the context of frames and the chapter ends with a remark which once more exhibits the adequacy of Weil entourages to bring out the meaning, in the context of frames, of the spatial results formulated in terms of entourages.
At the end of each chapter there is a section with additional references and comments.
An appendix (on page 141) contains two diagrams which summarize the relations between the various categories of spaces and frames presented along the text. We also list categories (page 151), symbols (page 153) and definitions (page 157) used throughout.
Our concern in relating the concepts introduced here with the ones already existing in the literature as well as in motivating the ideas developed in frames with the spatial situations, justifies the extensive bibliography included.
Proofs of already known results will usually be ommited. We think that all results included in Chapters I, II, III and IV for which no reference is given are original. In general, choice principles as the Axiom of Choice or the Countable Dependent Axiom of Choice are used without mention.