On automatic homeomorphicity for transformation monoids

TL;DR

This paper investigates conditions under which certain transformation monoids, equipped with the topology of point-wise convergence, are automatically homeomorphic when isomorphic to endomorphism monoids of specific structures.

Contribution

It establishes automatic homeomorphicity properties for monoids of non-decreasing functions on rationals, non-expansive functions on the Urysohn space, and endomorphisms of the universal homogeneous poset.

Findings

Monoid of non-decreasing functions on rationals has automatic homeomorphicity.

Monoid of non-expansive functions on the Urysohn space has this property.

Endomorphism monoid of the universal homogeneous poset also exhibits automatic homeomorphicity.

Abstract

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.