Yet Another Generalization of The Notion of a Metric Space

TL;DR

This paper introduces a new generalization of metric spaces called $oldsymbol{ ext{ extsterling}-}$metric spaces, exploring their topological properties and product spaces, expanding the theoretical framework of metric generalizations.

Contribution

It proposes a novel $ ext{ extsterling}$-metric concept based on a generalized triangle inequality and analyzes the topological structures and product spaces derived from it.

Findings

Defined the $ ext{ extsterling}$-metric and its properties.

Provided examples of non-trivial $ ext{ extsterling}$-metrizable spaces.

Studied the product topology for $ ext{ extsterling}$-metrizable spaces.

Abstract

A generalization of the triangle inequality is introduced by a mapping similar to a t-conorm mapping. This generalization leads us to a notion for which we use the $\star$-metric terminology. We are interested in the topological space induced by a $\star$-metric. Considering some examples of non-trivial $\star$-metrizable topological spaces, we also study the product topology for a finite family of $\star$-metrizable topological spaces.