Weakenings of compactness and normality on Isbell-Mrówka spaces, Hyperspaces of Vietoris and Abelian groups

• Ano de defesa: 2022
• Autor(a) principal: Rodrigues, Vinicius de Oliveira
• Tipo de documento: Tese
• Tipo de acesso: Acesso aberto
• Idioma: eng
• Instituição de defesa: Biblioteca Digitais de Teses e Dissertações da USP
• Palavras-chave em Português: Combinatória infinita; Compacidade enumerável; Countably compactness; Espaços de Isbell-Mrówka; General topology; Grupos topológicos; Hiperespaços de Vietoris; Hyperspaces of Vietoris; Infinitary combinatorics; Isbell-Mrówka spaces; Pseudocompacidade; Pseudocompacity; Topologia geral; Topological groups
• Link de acesso: https://www.teses.usp.br/teses/disponiveis/45/45131/tde-14062022-164023/

We provide an example of a Tychonoff almost-normal topological space which is not normal and explore almost-normality in the realm of Isbell-Mrówka spaces. Following this line, we study strongly aleph_0-separated almost disjoint families by comparing them with what is known about normal and pseudonormal almost disjoint families. We define a new family of special sets of reals related to these problems which we called weak lambda-sets. This study explores some questions of Paul Szeptycki and Sergio García-Balan. We explore John Ginsburg\'s questions on pseudocompact and countably compact Vietoris hyperspaces. In particular, we provide an example of a subspace of beta omega containing omega whose every power below the cardinal characteristic h is countably compact, but whose Vietoris hyperspace fails to be pseudocompact. We explore the converse implications in this class of spaces. We also study these questions in the realm of Isbell-Mrówka spaces, proving that the existence of a MAD family whose Vietoris hyperspace of its Isbell-Mrówka space is not pseudocompact is equivalent to the Baire number of omega* being less or equal to c. We also provide a consistent example of such an Isbell-Mrówka space of cardinality omega_2<c. Finally, we force a classification of non-torsion Abelian groups of size <= 2^c that admit a Hausdorff countably compact group topology containing convergent sequences, partially answering a question of Dikranjan and Shakhmatov.