Abstract
Function Clones (Clones) over a set $D$ is a set of functions of finite arity on $D$ which is closed under composition and contains the projections. Clones generalize transformation monoids. Clones and transformation monoids carry a natural topology, provided by the topology of pointwise convergence. A clone $\mathrm{Pol} \left( \mathcal{A} \right)$ has automatic homeomorphicity if every clone isomorphism from $\mathrm{Pol}\left(\mathcal{A}\right)$ to $\mathrm{Pol} \left(\mathcal{B}\right)$ is a homeomorphism. The definition of automatic homeomorphicity first appeared in connection with the reconstruction of the topology of clones, studied by Bodirsky, Pinsker and Pongrácz (2014). In this talk, we present the results obtained related to the automatic homeomorphicity of $\mathrm{Pol}\left(\mathbb{Q},\leq \right)$.