Master's Dissertation
DOI
https://doi.org/10.11606/D.45.2018.tde-26042018-162455
Document
Author
Full name
Rodrigo Rey Carvalho
E-mail
Institute/School/College
Knowledge Area
Date of Defense
Published
São Paulo, 2018
Supervisor
Committee
Junqueira, Lucia Renato (President)
Aurichi, Leandro Fiorini
Passos, Marcelo Dias
Title in Portuguese
O problema de Scarborough-Stone
Keywords in Portuguese
Compacidade
Pequenos cardinais
Problema de Scarborough-Stone
Abstract in Portuguese
O problema de Scarborough-Stone consiste em perguntar se o produto de espaços topológicos sequencialmente compactos precisa ser enumeravelmente compacto. Nesse trabalho estudamos alguns resultados que surgiram tentando resolver tal problema. Começamos com uma resposta negativa em ZFC usando espaços T2 e depois especificamos melhor condições sobre os axiomas de separação envolvendo os espaços do produto. Veremos respostas positivas envolvendo alguns axiomas de separação mais fortes como T6 (usando MA e a negação de CH) e T5 (usando o PFA). Além disso construÃmos mais respostas negativas usando construções como a Reta de Ostaszewski, espaços de Franklin-Rajagopalan e estruturas envolvendo álgebras Booleanas.
Title in English
The Scarborough-Stone problem
Keywords in English
Compactness
Scarborough-Stone problem
Small cardinals
Abstract in English
The Scarborough-Stone problem asks if every product of sequentially compact spaces must be a countably compact space. In this work we study some results that have arisen in attempt to solve this problem. We start our results with a negative answer in ZFC using T2 spaces and specify our conditions about the separability axioms of the spaces of the product. We will see positive answers assuming stronger separability axioms like T6 (using MA and the negation of CH) and T5 (using the PFA). We also construct more negative answers using constructions like the Ostaszewski line, Franklin-Rajagopalan spaces and structures involving Boolean algebras.
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Publishing Date
2018-11-23