Metrizability of Topologies and Generalized Distances
Keywords: Metrizable Topology, Generalized Metrics, Open Balls.
This paper presents 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 through the open balls. In addition, several interesting examples of topologies are
presented to show that several conditions 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 metric concept that also generates a topology and assess whether this topology is
metrizable.