Quantum isometries and loose embeddings

We show that countable metric spaces always have quantum isometry groups, thus extending the class of metric spaces known to possess such universal quantum-group actions. Motivated by this existence problem we define and study the notion of loose embeddability of a metric space (X,d_X) into another, (Y,d_Y): the existence of an injective continuous map that preserves both equalities and inequalities of distances. We show that 0-dimensional compact metric spaces are “generically” loosely embeddable into the real line, even though not even all countable metric spaces are.