Refining connected topological group topologies on Abelian torsion-free groups

Tkačenko, Michael G.; Villegas-Silva, Luis Miguel

doi:10.1016/s0022-4049(96)00109-0

Cited by 5 publications

(3 citation statements)

References 8 publications

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“…We answer Question 1 in the negative, even if S is a dense subgroup of D. To present a counterexample, we need the following result (see [12] (Lemma 4.7)): Proposition 3. Let G be an uncountable separable topological abelian group such that the torsion subgroup of G is countable.…”

Section: Proofmentioning

confidence: 99%

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

Tkachenko

2020

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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.

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Section: Proofmentioning

confidence: 99%

“…In fact, Proposition 3 is slightly more general than Lemma 4.7 in [12] since G is assumed to be torsion-free there. However, almost the same argument works under the conditions of Proposition 3 as well.…”

Section: Proofmentioning

confidence: 99%

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

Tkachenko

2020

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“…The general problem of finding a finer topological group topology on a topological abelian group G preserving some of the properties of G was considered by several authors in [1,[3][4][5][6][7]11] and [14,15,17,18]. This study has been mainly focused on the preservation of pseudocompactness or connectedness.…”

Section: Introductionmentioning

confidence: 99%