Continuity and Algebraic Structure of the Urysohn space

TL;DR

This paper provides a new constructive approach to the Urysohn space, enabling continuous isometry extensions and algebraic structure without choice assumptions, enhancing understanding of its foundational properties.

Contribution

It introduces an alternative, constructive construction of the Urysohn space that supports canonical isometry extensions and algebraic structure without relying on choice principles.

Findings

Isometries can be extended continuously in the new construction.

The Urysohn space can be equipped with algebraic structure constructively.

The approach avoids non-constructive choice assumptions.

Abstract

The Urysohn space is a complete separable metric space, universal among separable metric spaces for extending finite partial isometries into it. We present an alternative construction of the Urysohn space which enables us to show that extending isometries can be done in a canonical and continuous way, and allows us to equip the Urysohn space with algebraic structure. This is achieved in a constructive setting without assuming any choice principles.