Metrizabilidade de topologias e distâncias generalizadas

• Ano de defesa: 2020
• Autor(a) principal: Nascimento, Bismark Gonçalves do
• Tipo de documento: Dissertação
• Tipo de acesso: Acesso aberto
• Idioma: por
• Instituição de defesa: Brasil; UFRN; PROGRAMA DE PÓS-GRADUAÇÃO EM MATEMÁTICA APLICADA E ESTATÍSTICA
• Palavras-chave em Português: Topologia metrizável; i-Métrica V-Valorada; Topologia i-Quasi-Pseudome-trizável; Métrica intervalar; CNPQ::CIENCIAS EXATAS E DA TERRA::MATEMATICA
• Link de acesso: https://repositorio.ufrn.br/jspui/handle/123456789/28928

In this work, we present a study on the metrizability of topologies presenting the necessary conditions for a topology to be metrizable, i.e., it can be constructed starting from a metric originating from open balls. In addition, several interesting examples of topologies are presented to show that many of the presented are only necessary. Moreover, the Nagata-Smirnov Bing theorem is also mentioned, which presents necessary and sufficient conditions for a topology to be metrizable. In addition, we present a generalization of the concept of metric, which is called V-valued i-metric. Through this generalization we define the concepts of V-valued i-quasi-metric, V-valued i-pseudometric and V-valued iquasipseudometric and it is proved that every topology is i-quasi-pseudometrizable. Based on the theory of interval math an interval metric is constructed which is a particular case of i-metric. This interval metric also generates a topology and we assess whether this topology is metrizable.