  Every clone of functions comes naturally equipped with a topology---the
topology of pointwise convergence. A clone $\mathfrak{C}$ is said to have
automatic homeomorphicity with respect to a class $\mathcal{C}$ of clones, if
every clone-isomorphism of $\mathfrak{C}$ to a member of $\mathcal{C}$ is
already a homeomorphism (with respect to the topology of pointwise
convergence). In this paper we study automatic homeomorphicity-properties for
polymorphism clones of countable homogeneous relational structures. To this end
we introduce and utilize universal homogeneous polymorphisms. Next to two
generic criteria for the automatic homeomorphicity of the polymorphism clones
of free homogeneous structures we show that the polymorphism clone of the
generic poset with reflexive ordering has automatic homeomorphicity and that
the polymorphism clone of the generic poset with strict ordering has automatic
homeomorphicity with respect to countable $\omega$-categorical structures. Our
results extend and generalize previous results by Bodirsky, Pinsker, and
Pongr\'acz.
