$\tau$-metric spaces and convergence

Abstract

In this paper, based on the meaning of $\tau$-metric space we study the notion of convergence and ideal convergence on this field of spaces and investigate their
properties, comparing it also with the usual notions of convergence and ideal convergence on metric spaces. Especially, we study the meaning of convergence on $\tau$-metric spaces, giving new characterizations for this notion, new results for complete $\tau$-metric spaces and new notions of compactness and totally boundedness on such spaces. We also prove theorems that enrich a related theory. Finally, we insert and study the meaning of ideal convergence on $\tau$-metric spaces, giving also new characterizations and investigate its behavior under the view of classical meaning of ideal convergence.