  A compact subset K of the complex plane C is a set of polynomial
(respectively rational) approximation if P(K) = A(K) (respectively R(K) =
A(K)), where P(K) (respectively R(K)) is the family of functions on K which are
uniform limits of polynomials (respectively rational functions, having no poles
on K) and A(K) is the family of continuous functions on K, which are
holomorphic on the interior of K. In the class of compact sets, the property of
being a set of polynomial approximation is easily seen to be invariant under
holomorphic motion. We show that this is no longer the case for rational
approximation. Secondly, we show that the Riemann Hypothesis holds if and only
if a certain map is a holomorphic motion.
