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

Abstract:

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 $T_1$ topologies; Polish semigroup topologies; and we formulate a notion of the Zariski topology for monoids and inverse 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 automatic continuity. We show that many well-known, and extensively studied, 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 $\operatorname {Inj}(\mathbb {N})$ consisting of the injective transformations of $\mathbb {N}$; and the monoid $C(2^{\mathbb {N}})$ of continuous functions on the Cantor set $2^{\mathbb {N}}$.

The monoid $\mathbb {N} ^\mathbb {N}$ can be equipped with the product topology, where the natural numbers $\mathbb {N}$ have the discrete topology; this topology is referred to as the pointwise topology. We show that the pointwise topology on $\mathbb {N} ^\mathbb {N}$, and its analogue on $P_{\mathbb {N}}$, is the unique Polish semigroup topology on these monoids. The compact-open topology is the unique Polish semigroup topology on $C(2 ^\mathbb {N})$, and on the monoid $C([0, 1] ^\mathbb {N})$ of continuous functions on the Hilbert cube $[0, 1] ^\mathbb {N}$. The symmetric inverse monoid $I_{\mathbb {N}}$ has at least 3 Polish semigroup topologies, but a unique Polish inverse semigroup topology. The full binary relation monoid $B_{\mathbb {N}}$ has no Polish semigroup topologies, nor do the partition monoids. At the other extreme, $\operatorname {Inj}(\mathbb {N})$ and the monoid $\operatorname {Surj}(\mathbb {N})$ of all surjective transformations of $\mathbb {N}$ each have infinitely many distinct Polish semigroup topologies.

We prove that the Zariski topologies on $\mathbb {N} ^\mathbb {N}$, $P_{\mathbb {N}}$, and $\operatorname {Inj}(\mathbb {N})$ coincide with the pointwise topology; and we characterise the Zariski topology on $B_{\mathbb {N}}$.

Along the way we provide many additional results relating to the Markov topology, the small index property for monoids, and topological embeddings of semigroups in $\mathbb {N}^{\mathbb {N}}$ and inverse monoids in $I_{\mathbb {N}}$.

Finally, the techniques developed in this paper to prove the results about monoids are applied to function clones. In particular, we show that: the full function clone has a unique Polish topology; the Horn clone, the polymorphism clones of the Cantor set and the countably infinite atomless Boolean algebra all have automatic continuity with respect to second countable function clone topologies.