  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.
