# On expansive mappings

## TL;DR

This paper explores the properties of expansive mappings on metric spaces, showing that certain conditions like total boundedness preserve isometry, and introduces anticontractions to characterize boundedness.

## Contribution

It extends known results by relaxing compactness to total boundedness and introduces anticontractions as a new way to characterize boundedness in expansive mappings.

## Key findings

- Total boundedness preserves isometry in expansive mappings.
- Counterexample shows the converse of the main result does not hold.
- Anticontractions are introduced to characterize boundedness.

## Abstract

When finding an original proof to a known result describing expansive mappings on compact metric spaces as surjective isometries, we reveal that relaxing the condition of compactness to total boundedness preserves the isometry property and nearly that of surjectivity. While a counterexample is found showing that the converse to the above descriptions do not hold, we are able to characterize boundedness in terms of specific expansions we call anticontractions.