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 T1 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 N^N; the full binary relation monoid B_N; the partial transformation monoid P_N; the symmetric inverse monoid I_N; the monoid Inj(N) consisting of the injective transformations of N; and the monoid C(2^N) of continuous functions on the Cantor set 2^N. The monoid N^N can be equipped with the product topology, where the natural numbers N have the discrete topology; this topology is referred to as the pointwise topology. We show that the pointwise topology on N^N, and its analogue on P_N, is the unique Polish semigroup topology on these monoids. The compact-open topology is the unique Polish semigroup topology on C(2^N), and on the monoid C([0,1]^N) of continuous functions on the Hilbert cube [0,1]^N. The symmetric inverse monoid I_N has at least 3 Polish semigroup topologies, but a unique Polish inverse semigroup topology. The full binary relation monoid B_N has no Polish semigroup topologies, nor do the partition monoids. At the other extreme, Inj(N) and the monoid Surj(N) of all surjective transformations of N each have infinitely many distinct Polish semigroup topologies. We prove that the Zariski topologies on N^N, P_N, and Inj(N) coincide with the pointwise topology; and we characterise the Zariski topology on B_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 N^N and inverse monoids in I_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.