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

Polish topologies on endomorphism monoids of relational structures

“…Lemma 3.2 (cf. Lemma 5.1 in [14]). Let X be an infinite set, and let S be a subsemigroup of X X such that S contains all of the constant transformations, and for every x ∈ X there exists f x ∈ S such that (x)f −1 x = {x} and (X)f x is finite.…”

“…Lemma 3.5 (cf. Lemma 5.3 in [14]). Let X be an infinite set and let S be a subsemigroup of X X such that for every a ∈ X there exist α, β, γ 1 , .…”

“…for every s ∈ S there exists f s , g s ∈ S and t s ∈ A such that s = f s t s g s and for every neighbourhood B of t s the set f s (B ∩ A)g s is a neighbourhood of s. It is shown in [14,Theorem 3.1] that if a Polish semigroup S has property X with respect to a Polish subgroup G, then the topology on S is maximal among the Polish semigroup topologies on S. We begin by establishing a sufficient condition, based on the existence of certain endomorphisms, for End(X) equipped with the pointwise topology to have property X with respect to the automorphism group Aut(X) of X. We then proceed to isolate certain model-theoretic properties of X that imply this condition, and in particular introduce the strong amalgamation property with homomorphism gluing (SAHG) in Definition 4.11. This property is a strengthening of the classical strong amalgamation property and can also be thought of as an "almost free amalgamation property".…”

“…It follows that the pointwise topology is the unique Polish group topology on Sym(N). Uniqueness of compatible topologies has been studied for many further groups and semigroups, and more general objects such as clones; see for example [2,4,6,11,12,14,17,22,23,24,25,26,27,28,29,31,40,41,42,43,45,47,48,49].…”

Categorically Closed Topological Groups

“…The T i S-nontopologizability of countable semigroups can be characterized in terms of Zariski topologies, which were studied in [12,14,15,16,17,19].…”