C- and C -quotients in Pointfree Topology

We generalize a major portion of the classical theory of Cand C embedded subspaces to pointfree topology, where the corresponding notions are frame Cand C -quotients. The central results characterize these quotients and generalize Urysohn’s Extension Theorem, among others. The proofs require calculations in CL, the archimedean f -ring of frame maps from the topology of the reals into the frame L. We give a number of applications of the central results. Contents 1. Introduction 2 2. Background 2 2.1. Frames 2 2.2. Uniform frames 6 2.3. Archimedean f -rings 7 3. Calculating in CL 10 3.1. The fundamental formula 11 3.2. Cozero facts 12 3.3. Division in CL 16 4. Completeness properties of CL 17 4.1. The uniform completeness of CL 17 4.2. The -completeness of CL 21 5. Cozero covers 25 5.1. Principal covers 25 5.2. Linked covers and towers 28 6. Complete separation 34 6.1. Complete separation in archimedean f -rings 34 6.2. Complete separation in frames 37 7. C-quotients and C -quotients 40 7.1. C -quotients 40 7.2. C-quotients 42 8. Applications 47 8.1. Maximal extensions of frame maps into dense quotients 47 Date: January 1, 2003. 1991 Mathematics Subject Classi…cation. Primary 06D22, 06F25; Secondary 54B30, 54G05, 54G10, 18B30. Key words and phrases. frame, C -embedded subspace, archimedean f -ring. File name: cextens3.tex. 1 2 RICHARD N. BALL AND JOANNE WALTERS-WAYLAND 8.2. Dense quotients of completely regular frames 50 8.3. Normality 55 8.4. Disconnectivity 56 References 61 1. Introduction Cand C embedded subspaces play a central role in general topology, and the corresponding frame quotients, here termed Cand C -quotients, are no less important to pointfree topology. Indeed, the characterizations of these quotients draw together many of the central strands of frame theory. But what the pointfree formulation adds to the classical theory is a remarkable combination of elegance of statement, simplicity of proof, and increase of extent. The central results are the frame characterizations of Cand C -quotients, Theorems 7.1.1 and 7.2.7. Sections 2-6 are ground-clearing and machinery-building for these results, and Section 8 consists of applications of them. Crucial to both theorems is the preservation of certain features of the cozero parts of the underlying frames. Covering properties also play an important role, and such properties naturally tie in closely with uniformities. In particular, the characterization of C-quotients leads to the study of what might be called the geometry of cozero covers. Because a good deal of machinery is required to tie together the long list of concepts upon which these embeddings impinge, we mention here only one particularly novel and interesting completeness feature of CL which emerged; we call it -completeness. In order to close the circles of ideas we found it necessary to employ the algebraic properties of CL as an archimedean f -ring. Roughly sixty percent of the arguments hinge on the algebra, even though a much smaller percentage of the results themselves mention it. This, of course, is squarely in the tradition of classical general topology, but it is a heretofore underused technique for the study of frames. And this may be the central thrust of our work: archimedean f -rings are just as important for understanding frames as they are for understanding general topology. Finally, it is a pleasure to record our gratitude to a generous and erudite referee, whose suggestions signi…cantly improved the exposition of these results. 2. Background In this section we brie‡y record the de…nitions and standard results necessary to read this article. Although all the results are well known, we include a few proofs to spare the uninitiated reader the trouble of recovering information which is sometimes di¢ cult to dig out of the literature. 2.1. Frames. A frame is a complete lattice L with top element> and bottom element ? satisfying the frame distributivity law a ^ _