On functions preserving products of certain classes of semimetric spaces

TL;DR

This paper extends the study of functions that preserve product structures from metric spaces to broader classes of spaces satisfying weaker triangle inequalities, exploring their properties and relationships with known functions.

Contribution

It broadens the scope of functions preserving product structures to include spaces with weaker triangle inequalities, providing new analogues and examples.

Findings

Extended functions to b-metric spaces.

Provided analogues of previous results.

Connected new functions with existing literature.

Abstract

In the paper we continue the research of Bors\'{i}k and Dobo\v{s} on functions which allow us to introduce a metric to the product of metric spaces. In this paper we extend their scope on broader class of spaces which usually fail to satisfy the triangle inequality, albeit they tend to satisfy some weaker form of this axiom. In particular, we examine the behavior of functions preserving $b$-metric inequality. We provide analogues of the results of Bors\'{i}k and Dobo\v{s}, adjusted to the new, broader setting. The results we obtained are illustrated with multitude of examples. Furthermore, the connections of newly introduced families of functions with the ones already known from the literature are investigated.