Metrizabilidade de topologias e distâncias generalizadas
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 th...
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| Formato: | Dissertação |
| Idioma: | pt_BR |
| Publicado em: |
Brasil
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| Endereço do item: | https://repositorio.ufrn.br/jspui/handle/123456789/28928 |
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| Resumo: | 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. |
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