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Master's Dissertation
DOI
https://doi.org/10.11606/D.45.2018.tde-17122018-161936
Document
Author
Full name
Vinicius de Oliveira Rodrigues
E-mail
Institute/School/College
Knowledge Area
Date of Defense
Published
São Paulo, 2017
Supervisor
Committee
Tomita, Artur Hideyuki (President)
Aurichi, Leandro Fiorini
Boero, Ana Carolina
Title in Portuguese
Almost disjoint families em topologia
Keywords in Portuguese
Almost disjoint families
Combinatória infinita
Espaços de Mrówka
Topologia geral
Abstract in Portuguese
Uma almost disjoint family é uma coleção infinita de subconjuntos infinitos de números naturais tal que a interseção de quaisquer dois de seus elementos distintos é finita. Almost disjoint families podem ser utilizadas para construir um espaço topológico associado chamado de Psi-espaços, também conhecido como espaços de Mrówka. As propriedades topológicas deste espaço topológico dependem das propriedades combinatórias da família que o deu origem, e estes espaços podem ser utilizados para responder perguntas sobre topologia geral, muitas vezes não inicialmente relacionadas com almost disjoint families ou seus respectivos espaços de Mrówka. Neste documento, exploramos diversas construções envolvendo estes objetos utilizando combinatória infinita e princípios combinatórios como diamante, Axioma de Martin e técnicas como Forcing e tratamos de problemas envolvendo compactificações de Stone-Cech, espaços sequenciais, a propriedade de Lindelöf em espaços de funções, hiperespaços de Vietoris, dentre outros. O primeiro capítulo contém diversos pré-requisitos necessários para a leitura desta dissertação a fim de torná-la o mais autocontida possível. O segundo capítulo introduz as almost disjoint families e seus Psi-espaços associados, provando diversas propriedades importantes. Os demais capítulos são independentes entre si e tratam de problemas de Topologia Geral que podem ser solucionados com estes conceitos, ou de problemas que derivam destes conceitos.
Title in English
Almost disjoint families in topology
Keywords in English
Almost disjoint families
General topology
Infinitary combinatorics
Mrówka spaces
Abstract in English
An almost disjoint family is an infinite collection of infinite subsets of natural numbers such that the intersection of any two of its elements is finite. Almost disjoint families may be used to construct an associated topological space called psi space, also know as Mrówka space. The topological properties of this topological space depends on the combinatorical properties of the family that originated it, and these spaces may be used to answer questions in general topology, many times initially unrelated to almost disjoint families or to their Mrówka spaces. In this document, we explore several constructions involving these objects by using infinitary combinatorics and combinatorical principles like diamond, Martin's Axiom, forcing techniques and we treat abour problems regardins Stone-Cech compactifications, sequencial spaces, the property of Lindelöf on spaces of functions, hyperspaces of Vietoris, among others. The first chapter contains several pre requirements that are neccessary to read this dissertation in order to make it as self contained as possible. The second chapter introduces almost disjoint families and their associated Psi spaces, proving several important properties. The following chapters are independent from each other and treat about problems on General Topology that may be solved by using these concepts, or about problems that arises from these concepts.
 
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dissert.pdf (1.19 Mbytes)
Publishing Date
2018-12-18
 
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