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In this paper we explore the extent to which the algebraic structure of amonoid $M$ determines the topologies on $M$ that are compatible with itsmultiplication. Specifically we study the notions of automatic continuity;minimal Hausdorff or Polish semigroup topologies; and we formulate a notion ofthe Zariski topology for monoids.  If $M$ is a topological monoid such that every homomorphism from $M$ to asecond countable topological monoid $N$ is continuous, then we say that $M$ has\emph{automatic continuity}. We show that many well-known monoids haveautomatic continuity with respect to a natural semigroup topology, namely: thefull transformation monoid $\mathbb{N}^\mathbb{N}$; the full binary relationmonoid $B_{\mathbb{N}}$; 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 itsanalogue on $P_{\mathbb{N}}$, are the unique Polish semigroup topologies onthese monoids. The compact-open topology is the unique Polish semigrouptopology on $C(2^\mathbb{N})$ and $C([0, 1]^\mathbb{N})$. There are at least 3Polish semigroup topologies on $I_{\mathbb{N}}$, but a unique Polish inversesemigroup topology. There are no Polish semigroup topologies $B_{\mathbb{N}}$nor on the partitions monoids. At the other extreme, Inj$(\mathbb{N})$ and themonoid Surj$(\mathbb{N})$ of all surjective functions on $\mathbb{N}$ each haveinfinitely many distinct Polish semigroup topologies. We prove that the Zariskitopologies on $\mathbb{N}^\mathbb{N}$, $P_{\mathbb{N}}$, and Inj$(\mathbb{N})$coincide with the pointwise topology; and we characterise the Zariski topologyon $B_{\mathbb{N}}$. In Section 7: clones.

Automatic continuity, unique Polish topologies, and Zariski topologies on monoids and clones