Topological size of the set of universal and ultrahomogeneous retractions on the Urysohn space

TL;DR

This paper analyzes the topological complexity of the set of universal, ultrahomogeneous 1-Lipschitz retractions on the Urysohn space, introducing new properties and topologies.

Contribution

It introduces a new extension property $(UR^*)$ and a pointwise retract topology to study the Borel complexity and density of these retractions.

Findings

Studied Borel complexity of $
$-Lipschitz retractions.

Established the equivalence of $(UR^*)$ with universality and ultrahomogeneity.

Analyzed the density of $
$-Lipschitz retractions in the space of all retractions.

Abstract

In this paper, we investigate the set $\mathcal{U}(\mathbb{U})$ of universal and ultrahomogeneous $1$-Lipschitz retractions acting on the Urysohn space as the subspace of the space $\mathcal{R}(\mathbb{U})$ of all $1-$Lipschitz retractions defined on the Urysohn space. Especially, we study Borel complexity and density $\mathcal{U}(\mathbb{U})$ in $\mathcal{R}(\mathbb{U}).$ In order to do that, we introduce a new extension property $(UR^*)$ that is equivalent to the universality and ultrahomogeneity of a retraction, and a new pointwise retract topology.