We show that for any co-amenable compact quantum group A=C(G) there exists a
unique compact Hausdorff topology on the set EA of isomorphism classes of
ergodic actions of G such that the following holds: for any continuous field of
ergodic actions of G over a locally compact Hausdorff space T the map T->EA
sending each t in T to the isomorphism class of the fibre at t is continuous if
and only if the function counting the multiplicity of gamma in each fibre is
continuous over T for every equivalence class gamma of irreducible unitary
representations of G. Generalizations for arbitrary compact quantum groups are
also obtained. In the case G is a compact group, the restriction of this
topology on the subset of isomorphism classes of ergodic actions of full
multiplicity coincides with the topology coming from the work of Landstad and
Wassermann. Podles spheres are shown to be continuous in the natural parameter
as ergodic actions of the quantum SU(2) group. When A is separable, we also
introduce a notion of regular quantum metric on G, and show how to use it to
induce a quantum metric on any ergodic action of G in the sense of Rieffel.
Furthermore, we introduce a quantum Gromov-Hausdorff distance between ergodic
actions and show that it induces the above topology.Comment: References and lemmas 5.7 and 5.8 added. To appear in JF