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Synthetic Topology and Constructive Metric Spaces University of Ljubljana Faculty of Mathematics and Physics Department of Mathematics Davorin Leˇsnik Synthetic Topology and Constructive Metric Spaces Doctoral thesis Advisor: izr. prof. dr. Andrej Bauer arXiv:2104.10399v1 [math.GN] 21 Apr 2021 Ljubljana, 2010 Univerza v Ljubljani Fakulteta za matematiko in fiziko Oddelek za matematiko Davorin Leˇsnik Sintetiˇcna topologija in konstruktivni metriˇcni prostori Doktorska disertacija Mentor: izr. prof. dr. Andrej Bauer Ljubljana, 2010 Abstract The thesis presents the subject of synthetic topology, especially with relation to metric spaces. A model of synthetic topology is a categorical model in which objects possess an intrinsic topology in a suitable sense, and all morphisms are continuous with regard to it. We redefine synthetic topology in order to incorporate closed sets, and several generalizations are made. Real numbers are reconstructed (to suit the new background) as open Dedekind cuts. An extensive theory is developed when metric and intrinsic topology match. In the end the results are examined in four specific models. Math. Subj. Class. (MSC 2010): 03F60, 18C50 Keywords: synthetic, topology, metric spaces, real numbers, constructive Povzetek Disertacija predstavi podroˇcje sintetiˇcne topologije, zlasti njeno povezavo z metriˇcnimi prostori. Model sintetiˇcne topologije je kategoriˇcni model, v katerem objekti posedujejo intrinziˇcno topologijo v ustreznem smislu, glede na katero so vsi morfizmi zvezni. Definicije in izreki sintetiˇcne topologije so posploˇseni, med drugim tako, da vkljuˇcujejo zaprte mnoˇzice. Realna ˇstevila so (zaradi sin- tetiˇcnega ozadja) rekonstruirana kot odprti Dedekindovi rezi. Obˇsirna teorija je razvita, kdaj se metriˇcna in intrinziˇcna topologija ujemata. Na koncu prouˇcimo dobljene rezultate v ˇstirih izbranih modelih. Math. Subj. Class. (MSC 2010): 03F60, 18C50 Kljuˇcne besede: sintetiˇcen, topologija, metriˇcni prostori, realna ˇstevila, konstruktiven 8 Contents Introduction 11 0.1 Acknowledgements .................................. .. 11 0.2 Overview ......................................... 11 0.3 TerminologyandNotation.............................. .. 14 1 Constructive Setting 19 1.1 Intuitionistic Logic . 19 1.2 Topoi ........................................... 25 2 Synthetic Topology 31 2.1 FromPoint-SettoSyntheticTopology . ..... 31 2.2 Redefinition of Basic Notions in Synthetic Topology . ..... 45 2.3 (Co)limitsandBases.................................. 57 2.4 PredicativeSetting.................................. .. 62 3 Real Numbers and Metric Spaces 69 3.1 RealNumbers...................................... 69 3.2 MetricSpaces...................................... 87 3.3 MetricCompletions ................................... 95 4 Metrization Theorems 107 4.1 The WSO Principle ................................... 110 4.2 Transfer of Metrization via Quotients . ...... 121 4.3 Transfer of Metrization via Embeddings . ..... 128 4.4 ConsequencesofMetrization . .... 144 5 Models of Synthetic Topology 145 5.1 ClassicalMathematics ................................ 145 5.2 Number Realizability or Russian Constructivism . .... 146 5.3 Function Realizability or Brouwer’s Intuitionism . .... 153 5.4 GrosTopos ........................................ 153 Bibliography 161 Povzetek v slovenskem jeziku 167 10 Introduction 0.1 Acknowledgements I would like to thank my parents for all the support over the years. This gratitude extends to my other relatives and all my friends, too numerous to mention. I am grateful to the government of the Republic of Slovenia for funding my doctoral study, and to the staff at the Institute of mathematics, physics and mechanics for the employment and all the cooperation over the years. I thank my roommates Sara, Samo and Vesna (not to mention the two cats Odi and Edi) for being closest to me while I was writing this dissertation (as well as having to endure my crankiness during this time). Regarding the mathematical work for the thesis, I am grateful for insights and discussions with Paul Taylor, Matija Pretnar and Mart´ınEscard´o. Matija, my former schoolmate, was the one who introduced me to synthetic topology, and has been a good friend and collaborator for years. Mart´ınwas the first to explicitly define synthetic topology, and did good work on the subject; I am thankful for his cooperation and for hosting me in Birmingham during the finishing stage of the thesis. Most of all I would like to thank my advisor Andrej Bauer who singlehandedly taught me category theory, constructive mathematics and computability theory, has been extremely friendly, supportive and helpful, has taken considerable amount of time for discussions with me, introduced me to professional mathematics and professional mathematicians, and for being so patient with me, a trait which was no doubt tested numerous times. I am especially grateful for his trust in me, and for the freedom he has given me that I needed to finish my work. In his own thesis he wrote that he hoped to one day be as good an advisor to his students as his had been to him; I daresay he succeeded. 0.2 Overview In this introductory section we explain the title and give an overview of the thesis. We start by explaining the first word in the title. The ‘synthetic’ approach to mathematics started some three decades ago with synthetic differential geometry [32, 35, 53]. The differentials were originally thought of as “infinitely small” quantities (infinitesimals) until it was shown that no such real numbers (aside from the trivial case of 0) exist. Consequently a differential (form) nowadays means a section of a cotangent bundle, a distinctly more complicated object. In spite of this, reasoning with differentials as infinitesimal quantities still yields correct theorems and formulae, physicists get numerically correct results, and even mathematicians prefer this heuristic when dealing with differential equations (e.g. when solving them via separation of variables). Is there a way then to explain this, and develop mathematics of infinitely small quantities formally? One way to do it is to change the notion of real numbers; see e.g. Robinson’s non-standard anal- ysis [49]. This method works, but has an unfortunate consequence that even though differentials by themselves become simpler (compared to the classical approach), the entire analysis becomes more complicated since the real numbers are now more complicated objects. 12 An ideal solution would be if the real numbers stayed what they are (or at least altered in a way small enough as to not make them more complicated), and meaningful differentials still existed. As mentioned, their non-existence can in that case be proven, but the crucial assumption is the decidable equality of the reals, i.e. for all real numbers x, y ∈ R, either x = y or x 6= y. This is a special instance of the more general law of excluded middle LEM which states that for all propositions (more precisely, their truth values) p either p or the negation ¬p holds (“every statement is either true or false”). The corollary is that if we want non-trivial infinitesimals to be real numbers, we need to relinquish the use of decidability of equality on the reals, and consequently the use of the full LEM as well, leading us into the realm of constructive mathematics. Models of a theory in which we have the reals with infinitesimals can be constructed, showing that this theory is consistent. These models are complicated though (e.g. certain kind of sheaves over differential manifolds); what makes this approach useful is that we can forget about specific models, reasoning instead with a different kind of axioms and logical rules of inference, but within those working just like in the usual set theory (the new axioms essentially capture the existence and uniqueness of a derivative of every map from reals to reals). The point is that when studying a certain structure (smoothness in this case), the classi- cal approach is to equip sets with this additional structure and identify maps which preserve it, e.g. groups and group homomorphisms, vector spaces and linear maps, partial order and mono- tone maps, topological spaces and continuous maps, measure spaces and measurable maps, smooth manifolds and smooth maps etc. The fact that the structure is an addition onto a set, not a part of its intrinsic structure, brings with itself a lot of baggage — it is not sufficient to construct a space, one must also equip it with smooth structure (consider for example what it takes to fully define the tangent bundle), and after constructing a map, one has to verify that it is smooth (or if not, “smoothen” it, i.e. suitably approximate it with a smooth map). However, within a model of synthetic differential geometry every object (satisfying a certain microlinearity property) automatically has a unique smooth structure, and every map is automatically differentiable. Because the structure we study is ingrained in the sets and maps themselves, constructions and proofs simplify significantly.1 Still, by itself this merely means that it is easier to prove a statement within a synthetic model, but we don’t know yet whether that implies anything for the classical mathematics, so part of the synthetic approach is to also construct a bridge between them over which synthetic proofs translate to classical ones. In practice this usually means that a classical category is embedded into a synthetic model in such a way that content of logical statements is (sufficiently) preserved, and thus a synthetic theorem yields a corresponding classical
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