Mappings between distance sets or spaces Vom Fachbereich Mathematik der Universität Hannover zur Erlangung des Grades Doktor der Naturwissenschaften Dr. rer. nat. genehmigte Dissertation von Dipl.-Math. Jobst Heitzig geboren am 17. Juli 1972 in Hannover 2003 diss.tex; 17/02/2003; 8:34; p.1 Referent: Prof. Dr. Marcel Erné Korreferent: Prof. Dr. Hans-Peter Künzi Tag der Promotion: 16. Mai 2002 diss.tex; 17/02/2003; 8:34; p.2 Zusammenfassung Die vorliegende Arbeit hat drei Ziele: Distanzfunktionen als wichtiges Werkzeug der allgemeinen Topologie wiedereinzuführen; den Gebrauch von Distanzfunktionen auf den verschiedensten mathematischen Objekten und eine Denkweise in Begriffen der Abstandstheorie anzuregen; und schliesslich spezifischere Beiträge zu leisten durch die Charakterisierung wichtiger Klassen von Abbildungen und die Verallgemeinerung einiger topologischer Sätze. Zunächst werden die Konzepte des ,Formelerhalts‘ und der ,Übersetzung von Abständen‘ benutzt, um interessante ,nicht-topologische‘ Klassen von Abbildungen zu finden, was zur Charakterisierung vieler bekannter Arten von Abbildungen mithilfe von Abstandsfunktionen führt. Nachdem dann eine ,kanonische‘ Methode zur Konstruktion von Distanzfunktionen angegeben wird, entwickele ich einen geeigneten Begriff von ,Distanzräumen‘, der allgemein genug ist, um die meisten topologischen Strukturen induzieren zu können. Sodann werden gewisse Zusammenhänge zwischen einigen Arten von Abbildungen bewiesen, wie z. B. dem neuen Konzept ,streng gleichmässiger Stetigkeit‘. Es folgt eine neuartige Charakterisierung der Ähnlichkeitsabbildungen zwis- chen Euklidischen Räumen. Die Dissertation schliesst mit einigen Verallgemeinerungen bekannter Vervollständigungskonstruktionen und wichtiger Fixpunktsätze, und einer kurzen Studie über Techniken der Visualisierung von Abständen. Abstract The aim of this thesis is threefold: to reinstate distance functions as a principal tool of general topology; to promote the use of distance functions on various mathematical objects and a thinking in terms of distances also in non-topological contexts; and to make more specific contributions by characterizing important classes of mappings and generalizing some important topological results. I start by using the key concepts of ‘preservation of formulae' and ‘translation of distances' to extract interesting ‘non-topological’ classes of mappings, which leads to the characterization of many well-known types of mappings in terms of distance functions. After giving a ‘canonical’ method for constructing distance functions, a suitable notion of ‘distance spaces' will be developed, general enough to induce most topological structures. Then certain relationships between many kinds of mappings are proved, including the new concept of ‘strong uniform continuity', followed by a new characterization of the similarity maps between Euclidean spaces. The thesis closes with some generalizations of completions and fixed point theorems, and a short, self-contained study of distance visualization techniques. Schlagworte: Distanzfunktion, Abbildung, Topologie Key words: distance function, mapping, topology diss.tex; 17/02/2003; 8:34; p.3 Danksagung Dank geht an Herrn Dipl.-Math. Lars Ritter und Dr. Daniel Frohn für Diskussionen und gelegentliche Korrekturlesearbeiten, und an Dr. Jürgen Reinhold für die erfolgreiche gemeinsame Zählerei. Dr. Matthias Kriesell danke ich sehr für die Bereitschaft, sich immer wieder auch unausgegorene Ideen anzuhören und meinen Gehirnknoten weitere Kanten hinzuzufügen. Vor allen anderen gilt mein Dank meinem Doktorvater Prof. Dr. Marcel Tonio Erné, mit dem ich unzählbare Stunden fruchtbaren Gedankenaustauschs teilen durfte und dem es gelang, unermüdlich meinen mathematischen Horizont zu erweitern. Ohne seine Unterstützung wäre eine solche breitangelegte Studie wohl kaum möglich gewesen. diss.tex; 17/02/2003; 8:34; p.4 Contents A GENERAL CONCEPT OF DISTANCE 1. Distance sets 6 Definitions 6 Real distances 7 Multi-real distances 12 −− −− −− Distances in classical algebraic structures 15 Some other concepts of ‘generalized metric' 23 −− −− 2. Mappings 25 Distance sets with the same value monoid 25 Translating distances of different type 29 −− −− Comparing distance functions on the same set 40 −− BACK TO THE ROOTS OF TOPOLOGY 3. Convergence and closure 48 Converging to the right class of structures 48 Open and closed; filters and nets 54 −− −− Distances in point-free situations, and hyperspaces 62 −− 4. More on mappings 68 Topological properties of maps 68 Maps with both topological and non-topological properties 72 −− −− Finest distance structures 82 −− 5. Fundamental nets and completeness 96 Fundamental nets and Cauchy-filters 96 Notions of completeness 99 Distances between nets 102 −− −− −− Some completions 105 −− 6. Fixed points 110 Banach: Fixed points of Lipschitz-continuous maps 110 −− Sine–Soardi: Fixed points of contractive maps 114 Brouwer: Fixed points of continuous maps 115 −− −− APPENDIX Visualization of distances 122 Additional proofs 137 References 140 Indices 144 diss.tex; 17/02/2003; 8:34; p.5 diss.tex; 17/02/2003; 8:34; p.6 INTRODUCTION −−− Railroad, telephone, bicycle, automobile, air plane, and cinema revolutionized the sense of distance. [::: ] Distances depended on the effect of memory, the force of emotions, and the passage of time. Stephen Kern, The Culture Of Time And Space 1880–1918 In everyday language, ‘distance’ has always been something more general than the length of a segment in some geometrical space. Instead, the concept of ‘near’ and ‘far’ is one of the more important categories in human thinking. Extracting the abstract idea from the physical phenomenon, we speak of the growing distance to an old friend, of how near we are to reach a certain goal, or how far from being jealous. It is important that quite often the interesting question is not ‘‘how much is in between x and y'' but rather ‘‘how much is needed to get from x to y''. This somewhat dynamical interpretation of distance differs from the geometrical one in that it does not imply any symmetry, positivity, or strictness a priori. It is only natural when mathematicians, too, think of their objects as being related by the one or other kind of distance and how surprising is it that we still −− require mathematical distances to be real-valued, mostly symmetric, and non- negative? Before 1900, mathematical distances had beed used mainly in geometry and as a measure of difference between real numbers or functions. They had also played an important role for the clarification of the notion of ‘real number' itself, which in turn was a strong impetus for the development of topology. In the beginning of the last century, when Fréchet [Fré05, Fré06, Fré28] and Hausdorff [Hau14, Hau27, Hau49] initiated the axiomatic study of distances in the general setting of metric instead of geometric spaces, the real numbers were 1 2 therefore the natural candidates for the values of a distance function. Complex numbers or real vectors, being imaginable alternatives, would most certainly not have been considered suitable because of the difficulties in ordering such entities given that partial orders had not received much attention at that time. −− On the other hand, rational numbers had already long been known to be too special because of their lacking completeness. Just as in case of measure theory, it is therefore not surprising that the theoretical treatment of distances was dominated by a paradigm of using real numbers. Although, from the beginning, general topology was far more than the study of metric spaces, the question of which topological spaces can be endowed with a suitable metric, known as the ‘metrization problem', remained important. This was not only because metric spaces had very nice topological properties, mostly inherited from even nicer properties of the real numbers themselves, but also since the idea of distance remained a principal intuition in building new topological concepts, and because topological spaces alone had not enough structure to formulate certain interesting notions. For example, Lipschitz- and uniform continuity, or completeness, being of great importance in real analysis, cannot be expressed in terms of open sets alone. This motivated the search for suitable structural additives to general topo- logical spaces, which could well have led to an early study of substantially more general distance functions than real metrics. But despite only a few attempts in the latter direction, the researchers in this field soon focused on systems of subsets instead, ending up with the notion of ‘uniform space' (cf. [BHH98]). However, there were situations when distances had a great chance of being reconsidered passing virtually unnoticed. Van Dantzig [vD32], for −− instance, defined fundamental sequences in a topological group, using Menger's ‘Gruppenmetrik’ [Men31] without recognizing it as a distance function. Even more surprisingly, Kelley essentially proved that every uniformity (even every quasi-uniformity) comes from a family of real-valued distance functions [Kel55], but despite the popularity of his classical textbook, the theory of uniform spaces did not yet enter a possibly fruitful engagement with a theory of vector-valued metrics. The aim of this thesis is threefold: to reinstate distance functions as a principal tool of general topology; to promote the use of distance functions on various mathematical objects and a thinking in terms of distances also in non-topological contexts; and to make