On automatic homeomorphicity for transformation monoids

Abstract

© 2015, Springer-Verlag Wien. Transformation monoids carry a canonical topology—the topology of point-wise convergence. A closed transformation monoid M is said to have automatic homeomorphicity with respect to a class M of transformation monoids, if every monoid-isomorphism of M to a member of M is already a homeomorphism (with respect to the topology of pointwise convergence). In this paper we show automatic homeomorphicity-properties for the monoid of non-decreasing functions on the rationals, the endomorphism-monoid of the countable universal homogeneous poset, and the monoids of non-expansive self-maps on the rational Urysohn space and the rational Urysohn sphere, respectively.