  An ultrafilter $\mathcal{U}$ on a countable base {\em has continuous Tukey
reductions} if whenever an ultrafilter $\mathcal{V}$ is Tukey reducible to
$\mathcal{U}$, then every monotone cofinal map $f:\mathcal{U}\ra\mathcal{V}$ is
continuous when restricted to some cofinal subset of $\mathcal{U}$.
  In the first part of the paper, we give mild conditions under which the
property of having continuous Tukey reductions is inherited under Tukey
reducibility. In particular, if $\mathcal{U}$ is Tukey reducible to a p-point
then $\mathcal{U}$ has continuous Tukey reductions. In the second part, we show
that any countable iteration of Fubini products of p-points has Tukey
reductions which are continuous with respect to its topological Ramsey space of
$\vec{\mathcal{U}}$-trees.
