Metrizabilidade de topologias e distâncias generalizadas

Abstract

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.