Quantum isometries and loose embeddings

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