Transversal, T1-independent, and T1-complementary topologies

Błaszczyk, Aleksander; Tkachenko, Mikhail

doi:10.1016/j.topol.2017.08.018

Cited by 10 publications

(9 citation statements)

References 36 publications

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“…Since every infinite subgroup H of G is C ′ -cofinite and is not abelian by Proposition 5.4(2), we deduce by the above argument that H is not solvable. By Proposition 5.4 (5), an infinite non-abelian C ′ -cofinite group is torsion. Now we prove that G is bounded, of prime exponent bigger than 3.…”

Section: ′ -Cofinite Groupsmentioning

confidence: 98%

“…(3) is immediate and (4) follows from (3). (5). The group G cannot contain infinite cyclic subgroups H. Indeed, such an H is C ′ -cofinite, by (3).…”

Section: ′ -Cofinite Groupsmentioning

confidence: 99%

“…As H is obviously closed in C G , equation (5) shows that N G (H) is closed in C G too, so either N G (H) is finite, or N G (H) = G. In the second case, H is normal, so each of its elements h ∈ H has only finitely many conjugates in G, as they all lie in the finite group H. Then C G (h) has finite index in G, so that C G (h) = G and h ∈ Z(G).…”

Section: ′ -Cofinite Groupsmentioning

confidence: 99%

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Groups with cofinite Zariski topology and potential density

Bonatto¹,

Dikranjan²,

Toller³

2021

Preprint

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Tkachenko and Yaschenko [34] characterized the abelian groups G such that all proper unconditionally closed subsets of G are finite, these are precisely the abelian groups G having cofinite Zariski topology (they proved that such a G is either almost torsion-free or of prime exponent). The authors connected this fact to Markov's notion of potential density and the existence of pairs of independent group topologies. Inspired by their work, we examine the class C of groups having cofinite Zariski topology in the general case, obtaining a number of very strong restrictions on these groups in the non-abelian case which suggest the bold conjecture that a group with cofinite Zariski topology is necessarily either abelian or finite. We show that Tkachenko-Yaschenko theorem fails in the non-abelian case and we offer a natural counterpart in the general case using a partial Zariski topology and an appropriate stronger version of the property almost torsion-free.

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Section: ′ -Cofinite Groupsmentioning

confidence: 98%

“…(3) is immediate and (4) follows from (3). (5). The group G cannot contain infinite cyclic subgroups H. Indeed, such an H is C ′ -cofinite, by (3).…”

Section: ′ -Cofinite Groupsmentioning

confidence: 99%

Section: ′ -Cofinite Groupsmentioning

confidence: 99%

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Groups with cofinite Zariski topology and potential density

Bonatto¹,

Dikranjan²,

Toller³

2021

Preprint

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show abstract

“…Our first observation is that the group G 1 is not precompact -otherwise continuous characters of G 1 would separate elements of G 1 . Since every non-zero element of the countable group G 1 has order 2, one can apply [5,Theorem 5.28] to find an open neighborhood U of zero e 1 in G 1 and a (necessarily discontinuous) automorphism f of the group G 1 such that f (U) ∩ U = {e 1 }. In other words, the group G 1 is self-transversal.…”

Section: Proofmentioning

confidence: 99%

Factoring Continuous Characters Defined on Subgroups of Products of Topological Groups

Tkachenko¹

2021

Preprint

Self Cite

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We study factorization properties of continuous homomorphisms defined on subgroups (or submonoids) of products of (para)topological groups (or monoids). A typical result is the following one: Let $D=\prod_{i\in I}D_i$ be a product of paratopological groups, $S$ be a dense subgroup of $D$, and $\chi$ a continuous character of $S$. Then one can find a finite set $E\subset I$ and continuous characters $\chi_i$ of $D_i$, for $i\in E$, such that $\chi=\big(\prod_{i\in E} \chi_i\circ p_i\big)\hs1\res\hs1 S$, where $p_i\colon D\to D_i$ is the projection.

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“…Our first observation is that the group G 1 is not precompact -otherwise continuous characters of G 1 would separate elements of G 1 . Since every non-zero element of the countable group G 1 has order 2, one can apply ( [16], Theorem 5.28) to find an open neighborhood U of zero e 1 in G 1 and a (necessarily discontinuous) automorphism f of the group G 1 such that f (U) ∩ U = {e 1 }. In other words, the group G 1 is self-transversal.…”

Section: Proofmentioning

confidence: 99%

Factoring Continuous Characters Defined on Subgroups of Products of Topological Groups

Tkachenko¹

2021

Axioms

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This study is on the factorization properties of continuous homomorphisms defined on subgroups (or submonoids) of products of (para)topological groups (or monoids). A typical result is the following one: Let D=∏i∈IDi be a product of paratopological groups, S be a dense subgroup of D, and χ a continuous character of S. Then one can find a finite set E⊂I and continuous characters χi of Di, for i∈E, such that χ=∏i∈Eχi∘piS, where pi:D→Di is the projection.

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Transversal, T1-independent, and T1-complementary topologies

Cited by 10 publications

References 36 publications

Groups with cofinite Zariski topology and potential density

Groups with cofinite Zariski topology and potential density

Factoring Continuous Characters Defined on Subgroups of Products of Topological Groups

Factoring Continuous Characters Defined on Subgroups of Products of Topological Groups

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