Automatic continuity, unique Polish topologies, and Zariski topologies on monoids and clones
In this paper we explore the extent to which the algebraic structure of a monoid $M$ determines the topologies on $M$ that are compatible with its multiplication. Specifically we study the notions of automatic continuity; minimal Hausdorff or Polish semigroup topologies; and we formulate a notion of the Zariski topology for monoids. If $M$ is a topological monoid such that every homomorphism from $M$ to a second countable topological monoid $N$ is continuous, then we say that $M$ has \emph{automatic continuity}. We show that many well-known monoids have automatic continuity with respect to a natural semigroup topology, namely: the full transformation monoid $\mathbb{N}^\mathbb{N}$; the full binary relation monoid $B_{\mathbb{N}}$; the partial transformation monoid $P_{\mathbb{N}}$; the symmetric inverse monoid $I_{\mathbb{N}}$; the monoid Inj$(\mathbb{N})$ consisting of the injective functions on $\mathbb{N}$; and the monoid $C(2^{\mathbb{N}})$ of continuous functions on the Cantor set. We show that the pointwise topology on $\mathbb{N}^\mathbb{N}$, and its analogue on $P_{\mathbb{N}}$, are the unique Polish semigroup topologies on these monoids. The compact-open topology is the unique Polish semigroup topology on $C(2^\mathbb{N})$ and $C([0, 1]^\mathbb{N})$. There are at least 3 Polish semigroup topologies on $I_{\mathbb{N}}$, but a unique Polish inverse semigroup topology. There are no Polish semigroup topologies $B_{\mathbb{N}}$ nor on the partitions monoids. At the other extreme, Inj$(\mathbb{N})$ and the monoid Surj$(\mathbb{N})$ of all surjective functions on $\mathbb{N}$ each have infinitely many distinct Polish semigroup topologies. We prove that the Zariski topologies on $\mathbb{N}^\mathbb{N}$, $P_{\mathbb{N}}$, and Inj$(\mathbb{N})$ coincide with the pointwise topology; and we characterise the Zariski topology on $B_{\mathbb{N}}$. In Section 7: clones.