Apartness and Uniformity: A Constructive Development PDF
Preview Apartness and Uniformity: A Constructive Development
EE E Theory and Applications of Computability In cooperation with the association Computability in Europe Series Editors Prof. P. Bonizzoni Università Degli Studi di Milano-Bicocca Dipartimento di Informatica Sistemistica e Comunicazione (DISCo) 20126 Milan Italy [email protected] Prof. V. Brattka University of Cape Town Department of Mathematics and Applied Mathematics Rondebosch 7701 South Africa [email protected] Prof. S.B. Cooper University of Leeds Department of Pure Mathematics Leeds LS2 9JT UK [email protected] Prof. E. Mayordomo Universidad de Zaragoza Departamento de Informática e Ingeniería de Sistemas E-50018 Zaragoza Spain [email protected] Forfurther volumes: http://www.springer.com/series/8819 Books published in this series will be of interest to the research community and graduate students, with a unique focus on issues of computability. The perspective of the series is multidisciplinary, recapturing the spirit of Turing by linking theoretical and real-world concerns from computer science, mathematics, biology, physics, and the philosophy of science. The series includes research monographs, advanced and graduate texts, and books that offer an original and informative view of computability and computational paradigms. Series Advisory Board Samson Abramsky, University of Oxford Eric Allender, Rutgers, The State University of New Jersey Klaus Ambos-Spies, Universität Heidelberg Giorgio Ausiello, Università di Roma, “La Sapienza” Jeremy Avigad, Carnegie Mellon University Samuel R. Buss, University of California, San Diego Rodney G. Downey, Victoria University of Wellington Sergei S. Goncharov, Novosibirsk State University Peter Jeavons, University of Oxford Nataša Jonoska, University of South Florida, Tampa Ulrich Kohlenbach, Technische Universität Darmstadt Ming Li, University of Waterloo Wolfgang Maass, Technische Universität Graz Prakash Panangaden, McGill University Grzegorz Rozenberg, Leiden University and University of Colorado, Boulder Alan Selman, University at Buffalo, The State University of New York Wilfried Sieg, Carnegie Mellon University Jan van Leeuwen, Universiteit Utrecht Klaus Weihrauch, FernUniversität Hagen Philip Welch, University of Bristol Douglas S. Bridges • Luminiţa Simona Vîţă EE Apartness and Uniformity E A Constructive Development V Douglas S. Bridges Luminiţa Simona Vîţă Department of Mathematics Department of Mathematics and Statistics and Statistics University of Canterbury University of Canterbury Private Bag 4800 Private Bag 4800 Christchurch Christchurch New Zealand New Zealand [email protected] [email protected] ISSN2190-619X e-ISSN2190-6203 ISBN978-3-642-22414-0 e-ISBN978-3-642-22415-7 DOI10.1007/978-3-642-22415-7 Springer Heidelberg D ordrecht LondonNew York Library of Congress Control Number: 2011937283 ACM Compu ting Cla ssification (19 98): F.1 , F.4 AMS Codes: 03-XX, 68-XX © Springer-Verlag Berlin Heidelberg 2011 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broad- casting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: deblik, Berlin Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com) EPIGRAPH I may prefer my ideals. The fact that they may not survive is not a reason for not fighting for them. Schumpeter rightly said that people who believe that ideals have to be absolute are idolatrous barbarians. Civilization means that you must allow the possibility of change without ceasing to be totally dedicated to—and ready to die for—your ideals so long as you believe in them. Isaiah Berlin1 1FromConversationswithIsaiahBerlin (2nded.),byRaminJahanbegloo,HalbanPub- lishersLtd,London,2007. PREFACE The problem of finding a suitable constructive framework for general topology is important and elusive. Errett Bishop2 In the mid-1970s, Douglas Bridges came across the article [35], advocating the use of nearness in the teaching of first courses in analysis. This led him to consider a constructive theory based on proximity—or, rather, the construc- tivelymoreappropriateoppositenotionofapartness—asapossiblealternative approach to topology. Not, at that stage, being sufficiently mathematically experienced to make significant progress with this idea, he put it on one side until February 2000, when he and Lumini¸ta Vˆı¸t˘a began the project on apart- ness spaces that is discussed in the present monograph. Clearly, two heads were better than one, as we were able to initiate a project that, in the inter- veningyears,has grownsubstantially,withcontributionsfrommathematicians in several countries and continents. What do we mean by constructive in this context, and why might a con- structive approach to topology be interesting or significant? The answer to the first question is simple: by constructive mathematics we mean, roughly, mathematics with intuitionistic logic and a set- or type-theoretic foundation thatprecludesthederivationofthelawofexcludedmiddle;inotherwords,we mean mathematics carried out in the style of the late Errett Bishop [9]. Every proof that is constructive in this sense embodies an algorithm that can be, and in several cases has been, extracted and then implemented; further, the original proof shows that the implementable algorithm (if correctly extracted) 2LettertoDouglasBridges,dated14April1975. vii viii Preface meets its specifications. One other important feature of Bishop’s constructive mathematics is that it is completely consistent with classical mathematics— mathematics using classical logic. In regard to the second question, we recall Bishop’s deflationary remark [9] (page 63): Verylittleisleftofgeneraltopologyafterthatvehicleofclassicalmathe- maticshasbeentakenapartandreassembledconstructively.Withsome regret, plus a large measure of relief, we see this flamboyant engine collapse to constructive size. At the time he wrote [9], Bishop firmly believed that computation in analy- sis required uniform continuity—normally on compact sets—rather than the weaker notion of pointwise (let alone sequential) continuity: The concept of a pointwise continuous function is not relevant. A con- tinuousfunction[ontherealline]isonethatisuniformlycontinuouson compact intervals. ([9], pages ix–x) Since the theorem asserting the uniform continuity of every pointwise con- tinuous, real-valued mapping on the closed interval [0,1] cannot be proved- within Bishop-style constructive mathematics,3 general topology, essentially a pointwise matter, would therefore have little constructive relevance; what was needed was a framework (such as metric-space theory) in which uniform notions could be expressed. Bishop’s remark, taken with his suggestion, in Appendix A to [9], that con- structive theories like that of distributions might rely on ad hoc topological notions, may have served to deflect attention away from the problem of find- ing a suitable constructive framework for general topology. The advantage of solving this problem would presumably be that we would have a general, and hencealmostcertainlyclarifying,frameworkfortopologicalmatters.Moreover, workofIshihara[55]andothers,20yearsafterthepublicationof[9],hasshown that even sequential continuity has an important role in Bishop-style mathe- matics;sothereisdefinitelyaplaceforaconstructivetheoryofabstractspatial relationships that covers such highly non-uniform types of continuity. One can certainly tackle topology by simply constructivising the classical development of the subject from the usual three axioms about open sets.4 However, the theory of apartness that we present in this book does much more, by encompassing both point-set topology and the theory of uniform spaces. This, we believe, enables us to see those subjects in a clearer light. Wenowoutlinethecontentsofthethreechaptersinourbook,whichbegins with one that introduces informal constructive (intuitionistic) logic and set 3ThetheoremisprovableinonemodelofBishop-stylemathematics—namely,intuition- isticmathematics—butisfalseinthemodelinwhicheverythingisinterpretedrecursively. 4ThishasbeendonebyGrayson[46,47],Troelstra[87],andWaaldijk[91],thelasttwo workingwithinBrouwer’sfullintuitionisticframework. Preface ix theory, as well as—very briefly—the basic notions and notations for metric and topological spaces. In Chapter 2 we introduce axioms for a point-set apartness, a more com- putationally informative notion than nearness.5 We then explore some of the elementary consequences of our axioms, and examine the associated topology and various types of continuity of mappings. One interesting feature of our constructive approach is that intuitionistic logic reveals distinctions that are invisible to the classical eye. For example, we present three natural notions of continuity for maps between point-set apartness spaces, notions that coalesce classically but are quite distinct constructively. After dealing with continuity, the chapter continues with a discussion of convergencefornetsinanapartnessspace,followedbyastudyoftheapartness structure on the product of two apartness spaces. It ends with some remarks about impredicativity and how that phenomenon might be avoided. The theory really gets interesting—and a lot harder—when, in Chapter 3, we consider apartness between subsets. The five axioms on which this part of the theory is based enable us to bring most of the work on point-set apartness acrosstotheset-setenvironment.Everyset-setapartnessgivesrisetoapoint- setapartnessintheobviousway:apointxisapartfromasetS ifthesingleton subset {x} is apart from S. The canonical example of a set-set apartness arises from a quasi-uniform structure (which, if a certain symmetry condition holds, becomes a uniform structure).Havingintroducedbothapartnessandquasi-uniformspaces,weare well placed to examine the connection between, on the one hand, the uniform continuityofamappingf :X →Y betweenquasi-uniformspacesand,onthe other, strong continuity, in which if the two subsets of Y are apart, then their pre-images under f are apart in X. It is straightforward to show that uniform continuity entails strong continuity; but the converse direction of entailment is much harder to deal with. Similar difficulties are found in our discussion of the relation between Cauchy nets in the usual uniform-space sense, and nets that have the property of total Cauchyness relative to the apartness structure associated with a given uniform structure. (The corresponding property of total completeness of an apartness space X, whereby every totally Cauchy sequence inX hasa limitinX,is classicallyequivalent tothecompactness of the underlying apartness topology on X.) The difficulties, alluded to above, in the proofs of certain results are unified byourintroductionofageneralprooftechniquewhichhasseveralapplications inthetheoryandwhichresembles—butisconsiderablymorecomplexthan—a well-known technique introduced into constructive analysis by Ishihara [55]. Notonlydoesourtechniqueapplytothestudyofstronganduniformcontinu- ity,butalsoitleadstoaneatproofthat,undercertainconditions,theuniform 5Actually, our fundamental notion is that of a pre-apartness. An apartness is a pre- apartnesswithanextradisjunctionpropertythatenablesustosplitargumentsintocases.
Description:The list of books you might like
