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Master's Dissertation
DOI
10.11606/D.45.2015.tde-30082015-180119
Document
Author
Full name
Bartira Maués
E-mail
Institute/School/College
Knowledge Area
Date of Defense
Published
São Paulo, 2015
Supervisor
Committee
Junqueira, Lucia Renato (President)
Franco Filho, Antonio de Padua
Silva, Samuel Gomes da
Title in Portuguese
Uma introdução à Cp (X)
Keywords in Portuguese
Convergência pontual
Espaço das funções contínuas
l-equivalência e t-equivalência
Lindelöf em Cp
Teoremas de dualidade
Abstract in Portuguese
Neste trabalho estudamos algumas propriedades do espaço das funções contínuas munido da topologia da convergência pontual. Começamos estudando o espaço Cp(X) de forma geral, verificando que propriedades topológicas principais valem em Cp(X), usando teoremas de dualidade entre X e Cp(X). Em seguida estudamos a relação da estrutura topológica de X e a estrutura algébrica e topológica de Cp(X), onde o Teorema de Nagata é fundamental. Observamos algumas propriedades de X que são preservadas por l-equivalência ou t-equivalência, ou seja, que são determinadas pela estrutura linear topológica, ou pela estrutura topológica de Cp(X), respectivamente. Por último estudamos as condições para que Cp(X) seja um espaço de Lindelöf. Concluímos com a prova de Okunev de que o número de Lindelöf de Cp(X) é igual ao número de Lindelöf de Cp(X)xCp(X), para espaços fortemente zero-dimensionais X.
Title in English
An introduction on Cp(X)
Keywords in English
Duality theorems
l-equivalence and t-equivalence
Lindelöf in Cp
Pointwise convergence
Space of continuous functions
Abstract in English
In this work we study some properties of the space of continuous functions endowed with the topology of pointwise convergence. We begin by studying the space Cp(X) in general terms, verifying that the main topological properties are valid in Cp(X), using duality theorems between X and Cp(X). Next we study the relationship between the topological structure of X and the algebraic as well as topological structure of Cp(X), in which the Nagata theorem theorem is essential. We observe some properties of X, which are preserved by l-equivalence or t-equivalence, i.e., which are respectively determined either by the linear topological structure of Cp(X) or by its topological one. Finally we study in which conditions Cp(X) is a Lindelöf space. We conclude with the proof of Okunev that the Lindelöf number of Cp(X) is equal to the Lindelöf number of Cp(X)xCp(X), for strongly zero-dimensional spaces X.
 
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DissBartira.pdf (736.86 Kbytes)
Publishing Date
2015-08-31
 
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