A generalized metric-type structure with some applications

TL;DR

This paper introduces O-metric spaces, a broad generalization of metric spaces, exploring their topological properties, fixed point theorems, and applications in distance estimation, inequalities, and matrix optimization.

Contribution

It presents the novel concept of O-metric spaces, classifies metric types, and establishes fixed point results with applications in various mathematical contexts.

Findings

O-metric spaces generalize metric spaces and include most metric types.

A fixed point theorem for contractive-like maps in O-metric spaces is proved.

Applications include distance estimation, polygon inequalities, and matrix entry optimization.

Abstract

The paper introduces the class of O-metric spaces, a novel generalization of metric-type spaces, classifying almost all possible metric types into upward and downward O-metrics. We list some topologies arising from O-metrics and discuss convergence, sequential continuity, first countability and T$_2$ separation. The topology of an O-metric space can be generated by an upward O-metric on the space hence the focus on upward O-metric spaces. A theorem on the existence and uniqueness of a fixed point of some contractive-like map is proved and related with some other well known fixed point results in literature. Applications to the estimation of distances, polygon inequalities, and optimization of entries in infinite symmetric matrices are also given.