Continuous Homomorphisms Defined on (Dense) Submonoids of Products of Topological Monoids

Abstract

We study the factorization properties of continuous homomorphisms defined on a (dense) submonoid S of a Tychonoff product D = ∏ i∈I D i of topological or even topologized monoids. In a number of different situations, we establish that every continuous homomorphism f : S → K to a topological monoid (or group) K depends on at most finitely many coordinates. For example, this is the case if S is a subgroup of D and K is a first countable left topological group without small subgroups (i.e., K is an NSS group). A stronger conclusion is valid if S is a finitely retractable submonoid of D and K is a regular quasitopological NSS group of a countable pseudocharacter. In this case, every continuous homomorphism f of S to K has a finite type, which means that f admits a continuous factorization through a finite subproduct of D. A similar conclusion is obtained for continuous homomorphisms of submonoids (or subgroups) of products of topological monoids to Lie groups. Furthermore, we formulate a number of open problems intended to delimit the validity of our results.

Factoring Continuous Characters Defined on Subgroups of Products of Topological Groups

“…Proof. By Corollary 3.12 in [19], one can find a finite set E ⊂ I and a continuous character χ E of p E (S) such that χ = χ E • p E S, where p E : D → ∏ i∈E D i is the projection. By assumptions of the theorem, T = p E (S) is a dually embedded subgroup of D E = ∏ i∈E D i .…”

“…In Section 3 we complement several results from [19,Section 2] about continuous characters of a dense submonoid S of the P-modification of a product D = ∏ i∈I D i of topologized monoids. We show in Proposition 3 and Example 3 that if ϕ : S → H is a nontrivial continuous homomorphism of S to a topologized monoid of countable pseudocharacter, then the family J (χ) of the subsets J of the index set I such that ϕ depend on J is often a filter on I, and this filter can have empty intersection, even if S = D and the product D = Z(2) ω is a compact metrizable topological group (hence the P-modification of D is a discrete group).…”

“…The present article is a natural continuation of [18,19], where we study continuous homomorphisms of subgroups (submonoids) of products of topological groups (monoids). We establish there that in many cases, a continuous homomorphism f : S → H of a submonoid (subgroup) S of a product D = ∏ i∈I D i of topological monoids (groups) to a topological monoid (group) H admits a factorization in the form…”

“…If one can find a finite (countable) set J for which (1) holds true, we say that f has a finite (countable) type. Most of the results in [18,19] present different conditions on S and/or H under which f has a countable or even finite type. Purely algebraic aspects of this study can be found in [4].…”

“…Every Lie group is an NSS group. According to [19,Theorem 3.11], every continuous homomorphism of an arbitrary subgroup of a product of topological monoids to a Lie group has a finite type. This is an essential ingredient in several arguments presented in Section 2.…”