An Alternative Definition of the Completion of Metric Spaces

TL;DR

This paper introduces a novel categorical approach to defining the completion of metric spaces, offering an alternative to the classical dense property method and proving their equivalence.

Contribution

It presents a new definition of metric space completion rooted in category theory, generalizable to arbitrary categories, and establishes their equivalence with classical definitions.

Findings

New categorical definition of metric space completion

Proved equivalence with classical dense property approach

Potential for generalization to other categories

Abstract

In this article, the author proposes another way to define the completion of a metric space, which is different from the classical one via the dense property, and prove the equivalence between two definitions. This definition is based on considerations from category theory, and can be generalized to arbitrary categories.