Universal and homogeneous structures on the Urysohn and Gurarij spaces

Abstract

Using Fra\" iss\' e theoretic methods we enrich the Urysohn universal space by universal and homogeneous closed relations, retractions, closed subsets of the product of the Urysohn space itself and some fixed compact metric space, $L$-Lipschitz map to a fixed Polish metric space. The latter lifts to a universal linear operator of norm $L$ on the Lispchitz-free space of the Urysohn space. Moreover, we enrich the Gurarij space by a universal and homogeneous closed subspace and norm one projection onto a $1$-complemented subspace. We construct the Gurarij space by the classical Fra\" iss\' e theoretic approach.