On automatic homeomorphicity for transformation monoids

Transformation monoids carry a canonical topology --- the topology of point-wise convergence. A closed transformation monoid \(\mathfrak{M}\) is said to have automatic homeomorphicity with respect to a class \(\mathcal{K}\) of structures, if every monoid-isomorphism of \(\mathfrak{M}\) to the endomorphism monoid of a member of \(\mathcal{K}\) is automatically a homeomorphism. In this paper we show automatic homeomorphicity-properties for the monoid of non-decreasing functions on the rationals, the monoid of non-expansive functions on the Urysohn space and the endomorphism-monoid of the countable universal homogeneous poset.