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University of Cape Town Department of Mathematics and Applied Mathematics Faculty of Science The copyright of this thesis vests in the author. No quotation from it or information derived from it is to be published without full acknowledgementTown of the source. The thesis is to be used for private study or non- commercial research purposes only. Cape Published by the University ofof Cape Town (UCT) in terms of the non-exclusive license granted to UCT by the author. University University of Cape Town Department of Mathematics and Applied Mathematics Faculty of Science Convexity in quasi-metricTown spaces Cape ofby Olivier Olela Otafudu May 2012 University A thesis presented for the degree of Doctor of Philosophy prepared under the supervision of Professor Hans-Peter Albert KÄunzi. Abstract Over the last ¯fty years much progress has been made in the investigation of the hyperconvex hull of a metric space. In particular, Dress, Espinola, Isbell, Jawhari, Khamsi, Kirk, Misane, Pouzet published several articles concerning hyperconvex metric spaces. The principal aim of this thesis is to investigateTown the existence of an injective hull in the categories of T0-quasi-metric spaces and of T0-ultra-quasi- metric spaces with nonexpansive maps. Here several results obtained by others for the hyperconvex hull of a metric space haveCape been generalized by us in the case of quasi-metric spaces. In particular we haveof obtained some original results for the q-hyperconvex hull of a T0-quasi-metric space; for instance the q-hyperconvex hull of any totally bounded T0-quasi-metric space is joincompact. Also a construction of the ultra-quasi-metrically injective (= u-injective) hull of a T0-ultra-quasi- metric space is provided. Furthermore, we show that the u-injective hull of a totally bounded T0-ultra-quasi-metric space is joincompact. We prove that a commuting family of nonexpansive maps on a bounded q-hyperconvex T0-quasi- metric space intoUniversity itself has a common ¯xed point and the common ¯xed point set is q-hyperconvex. Some interesting examples related to the q-hyperconvex hull and u-injective hull are also discussed. i Acknowledgements I would like to thank my supervisor, Professor H.-P. KÄunzifor his in¯nite support, patience, proper guidance, stimulating discussions on the subject and encourage- ment given to me throughout my studies. Town I would also like to thank the Department of Mathematics and Applied Mathe- matics of the University of Cape Town in general, and in particular the Topology and Category Theory Research Group for allCape kinds of support and a marvelous research environment. of I would like to express my sincerest gratitude to the National Research Founda- tion (NRF) for providing a fund under Grant FA200622300009, for PhD Studies. Further ¯nancial assistance was provided by the University of Cape Town under an International Scholarship. I wish to thank the Laboratory of Mathematical Logic and Theoretical Computer Science of the UniversityUniversity of Siegen, which accepted me as a visitor in May 2011. To my friends Patrick Abedi, Joshua Bosange, Alphonse Kalala, Eric Madudu, Charly Makitu, Zachariah Mushaandja, and Jean-Marie Rusimbuka thank you for your support and encouragement. ii Dedication I dedicate this thesis to my parents Pauline Konge Esangowale and M¶edard Amundala Odimo, to my wife Nathalie Ntandunzadi, my boy Liouville Olela Ota- fudu and my girl Christella Olela Otafudu, for their patienceTown and understanding. To my family and friends, who are willing to carry out any scienti¯c work. Cape of University iii Declaration I, OLIVIER OLELA OTAFUDU hereby declare that this thesis is my own unaided work and is being submitted for the degree of Doctor of Philosophy at the UniversityTown of Cape Town. It has not been submitted for any degree or examination to any other university. Cape SIGNED:.............................. of DATE:................................ University iv Contents Abstract i Acknowledgements Town ii Dedication iii Cape Declaration of iv Introduction 1 0 Preliminaries 6 0.1 ConceptsUniversity of quasi-pseudometrics . 6 0.2 Concept of ultra-quasi-pseudometrics . 8 1 On hyperconvex metric spaces 12 1.1 Hyperconvexity . 12 v 1.2 Isbell's hyperconvex hull of a metric space . 14 1.3 The T -construction . 19 2 On ultra-metrically injective spaces 21 2.1 Spherical completeness . 22 2.2 Ultra-metrically tight extensions . 23 2.3 Ultra-metrically injective envelopes . 26 3 The q-hyperconvex hull of a T0-quasi-metric spaceTown 29 3.1 Convexity . 30 3.2 A space of nonnegative function pairs of the T0-quasi-metric space (X; d) .................................Cape 35 3.3 The q-hyperconvex hull ²q(X; dof) of a T0-quasi-metric space (X; d) 38 4 Properties of q-hyperconvex spaces 56 4.1 Intersection of q-hyperconvex spaces and ¯xed point theorems . 57 4.2 ApproximateUniversity ¯xed points . 65 4.3 External q-hyperconvexity . 71 5 The u-injective hull of a T0-ultra-quasi-metric space 75 5.1 Strongly tight function pairs . 75 vi 5.2 Envelopes or u-injective hulls of a T0-ultra-quasi-metric space . 80 5.3 q-spherical completeness . 83 5.4 Total boundedness in ultra-quasi-metric spaces . 98 6 Conclusion 103 6.1 Summary of the achieved work . 104 6.2 Two possible areas for future work . 106 Bibliography Town 108 Cape of University vii Introduction The concept of \hyperconvexity" for metric spaces was ¯rst introduced by Aron- szajn and Panitchpakdi [4](1956) who proved that a hyperconvex metric space is a nonexpansive absolute retract, i.e. it is a nonexpansive retract of any met- ric space in which it is isometrically embedded. The hyperconvexTown metric spaces called injective metric spaces were fully characterized by these authors. In par- ticular they pointed out the connection to the classical Hahn-Banach Theorem in normed spaces. As the term \hyperconvex"Cape suggests, this characterization in- volves a type of convexity which turns out to be stronger than metric convexity. Although this fact, and mainly the propertiesof that are deduced from this concept, might tell us that since hyperconvex spaces enjoy nice properties there might be a few such spaces, Isbell [29] has shown that every metric space has an envelope (injective hull) which is hyperconvex. In particular, a metric space and its com- pletion have exactly the same hyperconvex hull. The corresponding theory for normed linear spaces was developed by Gleason, Goodner, Kelley and Nachbin (see for instanceUniversity [44]). In the nonlinear theory interesting ¯xed point theorems were proved for nonexpansive mappings in bounded hyperconvex metric spaces [32]. Isbell [29], Dress [20], and Chrobak and Larmore [12] independently established that every metric space (X; d) has an injective hull, which is compact if X is compact (in a more general framework a similar result was presented in [32]). 1 Such a space is called the injective envelope (denoted by ²(X)) by Isbell, the convex hull by Chrobak and Larmore, and the maximal tight extension or tight span (denoted by TX ) by Dress. Since one of the most interesting concepts in the theory of hyperconvex metric spaces is the notion of the injective hull of a metric space [29], it is very natural to wonder about the existence of an injective hull in the ¯eld of generalized metric spaces. In this thesis we continued some investigations originally started by Salbany and Kemajou (a former student of my supervisor) on a reasonable concept of hyper- convexity in T0-quasi-metric spaces (compare [33]). It is proved by Salbany in [53] that a T0-quasi-metric space is q-hyperconvex if andTown only if it is metrically convex and q-hypercomplete. The main purpose of this thesis is to investigate the hyperconvex hull of a T0- quasi-metric space, which we call the q-hyperconvexCape hull of the T0-quasi-metric space. The main result is that the q-hyperconvex hull of a T0-quasi-metric space is unique up to isometries and we showof that the q-hyperconvex hull of any totally bounded T0-quasi-metric space is joincompact. Secondly we investigate the ultra- quasi-metrically injective hull of a T0-ultra-quasi-metric space in parallel to the q-hyperconvex hull of a T0-quasi-metric space. The main result is that a T0-ultra- quasi-metric space is ultra-quasi-metrically injective [which we call u-injective] if and only if is it q-spherically complete (see De¯nition 5.3.1). We present an explicit constructionUniversity of the u-injective hull of a T0-ultra-quasi-metric space and we also show that the u-injective hull of a totally bounded T0-ultra-quasi-metric space is joincompact. Note that the q-hyperconvex hull and the u-injective hull of an ultra-quasi-metric space need not coincide. And of course, we achieve all this by tackling several interesting questions, and as expected this leads us to some new open problems. The thesis is organized as described below. 2 A Brief Outline of The Thesis Chapter 0. In the ¯rst chapter we give a brief overview of certain well-known ba- sic concepts from the theory of quasi-pseudometric and ultra-quasi-pseudometric spaces. Some interesting examples of T0-quasi-metric spaces and T0-ultra-quasi- metric spaces (see Example 0.2.1) to be used throughout the thesis are presented. Chapter 1. In this chapter we collect the fundamental results about hypercon- vex metric spaces. We begin by de¯ning metrical convexity, hyperconvexity and metrical injectivity. We recall that a metric space is hyperconvex if and only if it is injective (see Theorem 1.1.1) and any hyperconvex space is complete (see Proposition 1.1.1). In the second section we present Isbell's ideas about the con- struction of the injective hull of a metric space.
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