  We prove that a compact quantum group with faithful Haar state which has a
faithful action on a compact space must be a Kac algebra, with bounded antipode
and the square of the antipode being identity. The main tool in proving this is
the theory of ergodic quantum group action on $C^*$ algebras.
  Using the above fact, we also formulate a definition of isometric action of a
compact quantum group on a compact metric space, generalizing the definition
given by Banica for finite metric spaces, and prove for certain special class
of metric spaces the existence of the universal object in the category of those
compact quantum groups which act isometrically and are `bigger' than the
classical isometry group.
