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

Rodrigues, Vinicius de Oliveira

doi:10.11606/T.45.2022.tde-14062022-164023

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DOI

https://doi.org/10.11606/T.45.2022.tde-14062022-164023

Document

Doctoral Thesis

Author

Rodrigues, Vinicius de Oliveira (Catálogo USP)

Full name

Vinicius de Oliveira Rodrigues

Institute/School/College

Instituto de Matemática e Estatística

Knowledge Area

Mathematics

Date of Defense

2022-06-03

Published

São Paulo, 2022

Supervisor

Tomita, Artur Hideyuki (Catálogo USP)

Committee

Tomita, Artur Hideyuki (President)
Aurichi, Leandro Fiorini
Boero, Ana Carolina
Silva, Samuel Gomes da
Szeptycki, Paul Jan

Title in English

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

Keywords in English

Countably compactness
General topology
Hyperspaces of Vietoris
Infinitary combinatorics
Isbell-Mrówka spaces
Pseudocompacity
Topological groups

Abstract in English

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

Title in Portuguese

Enfraquecimentos de compacidade e normalidade em espaços de Isbell-Mrówka, hiperespaços de Vietoris e grupos Abelianos

Keywords in Portuguese

Combinatória infinita
Compacidade enumerável
Espaços de Isbell-Mrówka
Grupos topológicos
Hiperespaços de Vietoris
Pseudocompacidade
Topologia geral

Abstract in Portuguese

Nós fornecemos um exemplo de espaço topológico Tychonoff, almost-normal não normal e exploramos almost-normalidade restrita aos espaços de Isbell-Mrówka. Seguindo essa linha de estudo, estudamos almost disjoint families fortemente aleph_0-separadas comparando elas ao que se sabe sobre almost disjoint families normais e pseudonormais. Definimos uma nova família de conjuntos especiais de números reais relacionadas a esses problemas que chamamos de weak lambda-sets. Esse estudo explora algumas questões de Paul Szeptycki e Sergio García-Balan. Nós exploramos as perguntas de John Ginsburg sobre pseudocompacidade e compacidade enumerável de hiperespaços de Vietoris. Em particular, obtivemos um exemplo de um subespaço de beta omega contendo omega cujas todas potências menores do que a característica cardinal h são enumeravelmente compactas, mas cujo hiperespaço de Vietoris não é pseudocompacto. Também exploramos essas perguntas restritas a espaços de Isbell-Mrówka, provando que a existência de uma MAD family cujo hiperespaço de Vietoris de seu espaço de Isbell-Mrówka não é pseudocompacto é equivalente ao número de Baire de omega* ser menor ou igual à c. Também obtivemos um exemplo consistente de um espaço de Isbell-Mrówka deste tipo de cardinalidade omega_2

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Publishing Date

2022-07-14

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