masterThesis
Metrizabilidade de topologias e distâncias generalizadas
Fecha
2020-03-02Registro en:
NASCIMENTO, Bismark Gonçalves do. Metrizabilidade de topologias e distâncias generalizadas. 2020. 85f. Dissertação (Mestrado em Matemática Aplicada e Estatística) - Centro de Ciências Exatas e da Terra, Universidade Federal do Rio Grande do Norte, Natal, 2020.
Autor
Nascimento, Bismark Gonçalves do
Resumen
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.