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 .................................
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