Almost disjoint families in topology

Author(s)

Vinicius de Oliveira Rodrigues

• Document type: Master's Dissertation
• Press: São Paulo.
• Institution: Universidade de São Paulo (USP). Instituto de Matemática e Estatística (IME/SBI)
• Defense date: 2017-12-11

Examining board members

Artur Hideyuki Tomita; Leandro Fiorini Aurichi; Ana Carolina Boero

• Advisor: Artur Hideyuki Tomita

Abstract

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. (AU)

• FAPESP's process: 15/15166-7 - M.A.D. families in topology
• Grantee: Vinicius de Oliveira Rodrigues
• Support Opportunities: Scholarships in Brazil - Master