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Master's Dissertation
DOI
10.11606/D.45.2007.tde-08052007-081304
Document
Author
Full name
Ana Carolina Boero
E-mail
Institute/School/College
Knowledge Area
Date of Defense
Published
São Paulo, 2007
Supervisor
Committee
Franco Filho, Antonio de Padua (President)
Alas, Ofelia Teresa
Silva, Samuel Gomes da
Title in Portuguese
Resolubilidade e irresolubilidade de espaços topológicos
Keywords in Portuguese
espaço d-forçado
irresolúvel
maximal
resolúvel
submaximal
Abstract in Portuguese
O principal objetivo deste trabalho é apresentar um estudo sistemático da teoria dos espaços topológicos resolúveis e irresolúveis. Enfocaremos diversas propriedades inerentes aos mesmos, incluindo uma exposição meticulosa de técnicas utilizadas na construção de espaços topológicos irresolúveis e sem pontos isolados. Dado um cardinal \kappa > 1, exibiremos exemplos de espaços topológicos que são \kappa-resolúveis, mas que não são \kappa^{+}-resolúveis. Mostraremos, ainda, que se um espaço topológico for n-resolúvel, para todo número natural n, o mesmo será \omega-resolúvel. Provaremos, contudo, que se \lambda é um cardinal tal que \omega < cf(\lambda) = \lambda, existe um espaço topológico que é \mu-resolúvel, para todo cardinal \mu < \lambda, mas que não é \lambda-resolúvel. O cerne desta dissertação refere-se à construção, em ZFC, de um subespaço enumerável, denso e submaximal de 2^c.
Title in English
Resolvable and irresolvable topological spaces
Keywords in English
d-forced space
irresolvable
maximal
resolvable
submaximal
topological space
Abstract in English
The main purpose of this work is to study the theory of resolvable and irresolvable topological spaces. We shall introduce many properties of these spaces and we shall give special attention to some techniques used in the construction of irresolvable topological spaces without isolated points. Given a cardinal \kappa > 1, we will present some examples of topological spaces which are \kappa-resolvable, but not \kappa^{+}-resolvable. Besides, we will show that if a topological space is n-resolvable, for every natural number n > 1, then it is \omega-resolvable too. Nevertheless, we shall prove that if \lambda is a cardinal with \omega < cf(\lambda) = \lambda, there is a topological space which is \mu-resolvable, for each cardinal \mu < \lambda, but that is not \lambda-resolvable. The backbone of this dissertation is the construction, in ZFC, of a countable, dense and submaximal subspaces of 2^c.
 
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Publishing Date
2007-06-25
 
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