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. Dr. Leandro Fiorini Aurichi USP – São Carlos Setembro de 2019 This work is dedicated to all those who do not restrain themselves by the finitude of our earthly lives and, occasionally, decide to venture into the infinite abstract. ACKNOWLEDGEMENTS I express my gratitude to FAPESP for the financial support to this project (2017/09797-0). I thank my advisor Leandro F. Aurichi and everyone from the topology group of ICMC (officially awarded as "The best group of 2018") for all the help throughout these years with the topology and set theory seminars. Here, I particularly thank Henrique A. Lecco, who made the question that motivated the beginning of paper [Aurichi and Duzi 2019]. My thanks also goes to Piotr Szewczak and Boaz Tsaban for giving us access to the preliminary notes of [Szewczak and Tsaban 2019]. Finally, I am specially thankful for the closest friends and family who have stood beside me through the ups and downs of this incredible journey. I would not dare to name all of you, but make no mistake: your support will never be forgotten. ABSTRACT DUZI, M. Topological games and selection principles. 2019. 166 p. Dissertação (Mestrado em Ciências – Matemática) – Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo, São Carlos – SP, 2019. This paper is dedicated to the beginning of the development of a book introducing topological games and selection principles. Here, the classical games (such as the Banach-Mazur) and selection principles (such as the Rothberger or Menger properties) are presented. The most notable applications are also displayed – both the classical (such as the characterization of Baire spaces with the Banach-Mazur game) and the recent (such as the relation between the Menger property and D-spaces). In addition to the content for the book, a problem in finite combinatorics that was found in the study of positional strategies is presented (as well as a partial solution) together with some results regarding new variations of classical selection principles and games, which give rise to the characterization of some notable spaces. Keywords: topological games, selection principles, tightness properties, covering properties, Baire spaces. RESUMO DUZI, M. Jogos topológicos e princípios seletivos. 2019. 166 p. Dissertação (Mestrado em Ci- ências – Matemática) – Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo, São Carlos – SP, 2019. Este trabalho é dedicado ao início do desenvolvimento de um livro introdutório à jogos to- pológicos e princípios seletivos. Aqui, são apresentados os clássicos jogos (tais como o de Banach-Mazur) e princípios seletivos (tais como a propriedade de Rothberger ou de Menger). Também são exibidas as aplicações mais notáveis encontradas na literatura – tanto as mais tradicionais (tais como a caracterização de espaços de Baire com o jogo de Banach-Mazur), como as mais atuais (tais como a relação entre a propriedade de Menger e D-espaços). Além do conteúdo voltado para o livro, são apresentados um problema de combinatória finita (assim como uma solução parcial para tal) que foi encontrado com o estudo de estratégias posicionais e alguns resultados envolvendo novas variações de princípios de seleção e jogos clássicos, possibilitando a caracterização de alguns espaços notáveis. Palavras-chave: jogos topológicos, princípios seletivos, propriedades de tightness, propriedades de coberturas, espaços de Baire. CONTENTS Introduction...................................... 15 I BOOK 19 1 PRELIMINARIES ............................ 21 1.1 Set theory .................................. 21 1.2 Topology ................................... 22 2 INTRODUCING GAMES........................ 27 2.1 Finite games ................................. 28 2.2 Infinite games ................................ 29 2.3 Strategies ................................... 30 2.4 Positional strategies ............................. 37 2.5 The Banach-Mazur game ......................... 42 3 SELECTION PRINCIPLES....................... 49 3.1 Classes of selection principles ....................... 49 3.2 Closure properties .............................. 51 3.3 Covering properties ............................. 52 3.3.1 Menger spaces ................................ 52 3.3.2 Rothberger spaces .............................. 53 3.3.3 Hurewicz spaces ............................... 55 3.3.4 Alster spaces ................................. 58 3.3.5 The Scheepers Diagram .......................... 59 4 THE ASSOCIATED SELECTIVE GAMES .............. 67 4.1 Covering games ............................... 69 4.1.1 The Rothberger game ........................... 69 4.1.2 The Menger game .............................. 72 4.1.3 The Pawlikowski Theorem ......................... 81 4.1.4 The Hurewicz game ............................. 87 4.1.5 The Alster game ............................... 87 4.2 Closure games ................................ 90 5 SOME CONNECTIONS AND APPLICATIONS ........... 97 5.1 The Banach-Mazur game and Baire spaces ............... 97 5.2 The Rothberger game and Measure Theory .............. 100 5.2.1 Strong measure zero ............................ 100 5.2.2 Purely atomic measures .......................... 102 5.3 Productively Lindelöf spaces ........................ 105 5.4 Sieve completeness ............................. 107 5.5 D-spaces ................................... 109 5.6 Tightness games and countable tightness ................ 114 5.7 The space of real continuous functions ................. 117 6 DIAGRAMS ...............................123 II RESULTS 127 7 POSITIONAL STRATEGIES AND A PROBLEM IN FINITE COM- BINATORICS ..............................129 8 TOPOLOGICAL GAMES OF BOUNDED SELECTIONS . 133 8.1 Selection Principles ............................. 134 8.2 The associated games ........................... 135 8.3 The “modfin” and “mod1” variations ................... 139 8.4 The analogous to Pawlikowski’s and Hurewicz’s results ....... 147 8.5 The dual game ................................ 151 8.6 Conclusion .................................. 153 Bibliography .....................................159 15 INTRODUCTION Games have been an object of study for mathematicians since, arguably, the 17th century. It was only in the 20th century that, in the Scottish Book, the game henceforth known as the first infinite mathematical game was introduced: the Banach-Mazur game (for a brief history of games in mathematics we refer to [Telgársky 1987] or [Aurichi and Dias 2019]). Obviously, infinite games are not meant to be played. But what good is a game, if not for playing it? Infinite games are used to describe certain infinite combinatorial properties in an intuitive manner. This description usually happens in terms of winning strategies. The property of being a Baire space, for instance, is characterized by one of the players not having a winning strategy in the Banach-Mazur game, while if the other player has a winning strategy in this game we can go even further and conclude that the space is productively Baire. Another way to describe combinatorial properties is through selection principles. In essence, these are properties asserting that one can choose an element from each set of a sequence in order to obtain a significant object. Selection principles define classical properties, such as Rothberger or Menger spaces. Our initial goal was to start the development of a book introducing these infinite games and selection principles in the context of topology. Midway through this project, however, we have obtained some new results on the field, so we decided to divide this text into two parts, each with its respective chapters: Part I This