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 withits 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 toa second countable topological monoid N is continuous, then we say that Mhas automatic continuity. We show that many well-known, and extensivelystudied, monoids have automatic continuity with respect to a natural semigroup topology, namely: the full transformation monoid T(N) on the natural numbers N; the full binary B(N)relation monoid; the partial transformation monoid P(N); the symmetric inverse monoid I(N); the monoid Inj(N) consisting of the injective transformationsof N; and the monoid Con(C) of continuous functions on the Cantor set C.The monoid T(N) can be equipped with the product topology, where the natural numbers N have the discrete topology; this topology is referred to as thepointwise topology. We show that the pointwise topology on T(N), and its analogue on P(N), is the unique Polish semigroup topology on these monoids. Thecompact-open topology is the unique Polish semigroup topology on Con(C), andon the monoid Con(H) of continuous functions on the Hilbert cube H.The symmetric inverse monoid I(N) has at least 3 Polish semigroup topologies, but a unique Polish inverse semigroup topology. The full binary relationmonoid 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 T(N), P(N), and Inj(N) coincide withthe pointwise topology; and we characterise the Zariski topology on B(N).Along the way we provide many additional results relating to the Markovtopology, the small index property for monoids, and topological embeddingsof semigroups in T(N) and inverse monoids in I(N).Finally, the techniques developed in this paper to prove the results aboutmonoids are applied to function clones. In particular, we show that: the fullfunction clone has a unique Polish topology; the Horn clone, the polymorphismclones of the Cantor set and the countably infinite atomless Boolean algebraall have automatic continuity with respect to second countable function clonetopologies.