Set Theory of Infinite Imperfect Information Games
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Dissertation.Pdf
Limited Information Strategies for Topological Games by Steven Clontz A dissertation submitted to the Graduate Faculty of Auburn University in partial fulfillment of the requirements for the Degree of Doctor of Philosophy Auburn, Alabama May 9, 2015 Keywords: topology, uniform spaces, infinite games, limited information strategies Copyright 2015 by Steven Clontz Approved by Gary Gruenhage, Chair, Professor Emeritus of Mathematics and Statistics Stewart Baldwin, Professor of Mathematics and Statistics Chris Rodger, Professor of Mathematics and Statistics Michel Smith, Professor of Mathematics and Statistics George Flowers, Dean of the Graduate School Abstract A topological game G(X) is a two-player game which characterizes properties of a topo- logical space X based upon the existence of winning perfect-information strategies for players in the game. If the property P is characterized by a player having a winning perfect infor- mation strategy, then a property stronger than P is characterized by that player having a winning limited information strategy. This paper investigates the existence of limited infor- mation strategies for four types of topological games from the literature, and the topological properties characterized by those strategies. ii Acknowledgments I'm very grateful for the support of all my friends and family during the completion of this manuscript, particularly for all the faculty members and graduate student colleagues I've worked with over the years at Auburn. I'd like especially to thank my adviser Gary Gru- enhage for all his support and insight on writing this paper and beginning my mathematical career. This dissertation is dedicated to my wife Jessica, for her love and encouragement. -
GAMES and SUPERTASKS Copia
Topological Games, Supertasks, and (Un)determined Experiments Thomas Mormann Abstract. The general aim of this paper is to introduce some ideas of the theory of infinite topological games into the philosophical debate on supertasks. First, we discuss the ele- mentary aspects of some infinite topological games, among them the Banach-Mazur game. Then it is shown that the Banach-Mazur game may be conceived as a Newtonian supertask. In section 4 we propose to conceive physical experiments as infinite games. This leads to the distinction between determined and undetermined experiments and the problem of how it is related to that between determinism and indeterminism. Finally the role of the Axiom of Choice as a source of indeterminacy of supertasks is discussed. Keywords: Infinite Topological Games, Banach Mazur Game, Supertasks, Newtonian Me- chanics, Axiom of Choice, Axiom of Determinacy, (Un)determined Experiments. 1. Supertasks in Physics and Mathematics 2. Topological Games 3. Newtonian Supertasks 4. Experiments as Games 5. The Axiom of Choice and the Axiom of Determinacy References 1. Supertasks in Physics and in Mathematics. A supertask (of order type ω) may be defined as an infinite sequence sn of actions sn, n = 1, 2, 3, … carried out one after another in a finite time. For instance, Achilles’s running a distance of 100 meters with constant speed may be considered as such a supertask: the first task s1 consists in running the first 50 meters in five seconds, the second task s2 in running the next 25 meters in 2.5 seconds, and so on, reaching the finish in exactly 10 seconds. -
Domain Representability and Topological Completeness
University of Dayton eCommons Honors Theses University Honors Program Spring 4-2016 Domain Representability and Topological Completeness Matthew D. DeVilbiss University of Dayton Follow this and additional works at: https://ecommons.udayton.edu/uhp_theses Part of the Geometry and Topology Commons eCommons Citation DeVilbiss, Matthew D., "Domain Representability and Topological Completeness" (2016). Honors Theses. 84. https://ecommons.udayton.edu/uhp_theses/84 This Honors Thesis is brought to you for free and open access by the University Honors Program at eCommons. It has been accepted for inclusion in Honors Theses by an authorized administrator of eCommons. For more information, please contact [email protected], [email protected]. Domain Representability and Topological Completeness Honors Thesis Matthew D. DeVilbiss Department: Mathematics Advisor: Lynne Yengulalp, Ph.D. April 2016 Domain Representability and Topological Completeness Honors Thesis Matthew D. DeVilbiss Department: Mathematics Advisor: Lynne Yengulalp, Ph.D. April 2016 Abstract Topological completeness properties seek to generalize the definition of complete metric space to the context of topologies. Chapter 1 gives an overview of some of these properties. Chapter 2 introduces domain theory, a field originally intended for use in theoretical computer science. Finally, Chapter 3 examines how this computer-scientific notion can be employed in the study of topological completeness in the form of domain representability. The connections between domain representability -
Generalized Polish Spaces at Regular Uncountable Cardinals 3
GENERALIZED POLISH SPACES AT REGULAR UNCOUNTABLE CARDINALS CLAUDIO AGOSTINI, LUCA MOTTO ROS, AND PHILIPP SCHLICHT Abstract. In the context of generalized descriptive set theory, we systemat- ically compare and analyze various notions of Polish-like spaces and standard κ-Borel spaces for κ an uncountable (regular) cardinal satisfying κ<κ = κ. As a result, we obtain a solid framework where one can develop the theory in full generality. We also provide natural characterizations of the generalized Cantor and Baire spaces. Some of the results obtained considerably extend previous work from [CS16, Gal19, LS15], and answer some questions contained therein. Contents 1. Introduction 1 2. Polish-like spaces 3 2.1. Definitions... 3 2.2. ...and their relationships 7 3. Characterizations of κκ andκ 2 22 4. Standard κ-Borel spaces 30 5. Final remarks and open questions 35 References 41 1. Introduction Generalized descriptive set theory is a very active field of research which in recent years received a lot of attention because of its deep connections with other areas such as model theory, determinacy and higher pointclasses from classical descriptive set theory, combinatorial set theory, classification of uncountable structures and non- arXiv:2107.02587v1 [math.LO] 6 Jul 2021 separable spaces, and so on. Basically, the idea is to replace ω with an uncountable cardinal in the definitions of the Baire space ωω and Cantor spaceω 2, as well as in all other topologically-related notions. For example, one considers κ-Borel sets (i.e. sets in the smallest κ+-algebra generated by the topology) instead of Borel sets, κ-Lindel¨of spaces (i.e. -
Universid Ade De Sã O Pa
Topological games and selection principles Matheus Duzi Ferreira Costa Dissertação de Mestrado do Programa de Pós-Graduação em Matemática (PPG-Mat) UNIVERSIDADE DE SÃO PAULO DE SÃO UNIVERSIDADE Instituto de Ciências Matemáticas e de Computação Instituto Matemáticas de Ciências SERVIÇO DE PÓS-GRADUAÇÃO DO ICMC-USP Data de Depósito: Assinatura: ______________________ Matheus Duzi Ferreira Costa Topological games and selection principles Dissertation submitted to the Institute of Mathematics and Computer Sciences – ICMC-USP – in accordance with the requirements of the Mathematics Graduate Program, for the degree of Master in Science. EXAMINATION BOARD PRESENTATION COPY Concentration Area: Mathematics Advisor: Prof. Dr. Leandro Fiorini Aurichi USP – São Carlos September 2019 Ficha catalográfica elaborada pela Biblioteca Prof. Achille Bassi e Seção Técnica de Informática, ICMC/USP, com os dados inseridos pelo(a) autor(a) Duzi, Matheus D812t Topological games and selection principles / Matheus Duzi; orientador Leandro Fiorini Aurichi. - - São Carlos, 2019. 166 p. Dissertação (Mestrado - Programa de Pós-Graduação em Matemática) -- Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo, 2019. 1. Topological games. 2. Selection principles. 3. Covering properties. 4. Tightness properties. 5. Baire spaces. I. Fiorini Aurichi, Leandro , orient. II. Título. Bibliotecários responsáveis pela estrutura de catalogação da publicação de acordo com a AACR2: Gláucia Maria Saia Cristianini - CRB - 8/4938 Juliana de Souza Moraes - CRB - 8/6176 Matheus Duzi Ferreira Costa Jogos topológicos e princípios seletivos Dissertação apresentada ao Instituto de Ciências Matemáticas e de Computação – ICMC-USP, como parte dos requisitos para obtenção do título de Mestre em Ciências – Matemática. EXEMPLAR DE DEFESA Área de Concentração: Matemática Orientador: Prof.