Contraction of a Generalized Metric Structure

TL;DR

This paper introduces galactic spaces, a new metric structure based on ordered field extensions, capable of modeling topological modifications via contractions, including infinitesimal ratios, with applications to nonstandard analysis.

Contribution

The paper develops galactic spaces as a novel formalism for representing topological changes, extending metric structures with contractions that include infinitesimal ratios.

Findings

Galactic spaces depend on ordered field extensions of R.

Contractions generalize homotheties and can have infinitesimal ratios.

An infinite family of galactic spaces can be associated to any metric space.

Abstract

In some scientific fields, a scaling is able to modify the topology of an observed object. Our goal in the present work is to introduce a new formalism adapted to the mathematical representation of this kind of phenomenon. To this end, we introduce a new metric structure - the galactic spaces - which depends on an ordered field extension of R. Moreover, some natural transformations of the category of galactic spaces, the contractions, are of particular interest: they generalize usual homotheties, they have a ratio which may be an infinitesimal, they are able to modify the topology and they satisfy a nice composition rule. With the help of nonstandard extensions we can associate to any metric space an infinite family of galactic spaces; lastly, we study some limit properties of this family.