Selected topics from the structure theory of topological groups

Dikranjan, Dikran; Shakhmatov, Dmitri

doi:10.1016/b978-044452208-5/50041-7

Cited by 25 publications

(46 citation statements)

References 36 publications

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“…The articles [3] (3.5, 5.9), [4] (Problems 2.2, 3.2), and [27] (Problem 9) provide some references and discussion.] Question 1.…”

Section: ) a Dense Subgroup Of A Pseudocompact Group Is Itself Pseudomentioning

confidence: 99%

Pseudocompact groups: progress and problems

Comfort

2008

Topology and its Applications

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Several months ago the speaker and Jan van Mill gave a proof of this result [W.W. Comfort, J. van Mill, Extremal pseudocompact abelian groups are compact metrizable, Abstracts Amer. Math. Soc. 27 (2006) 78 (Abstract #1014-22-958); W.W. Comfort, J. van Mill, Extremal pseudocompact abelian groups are compact metrizable, Proc. Amer. Math. Soc. 135 (2007) 4039-4044]:A pseudocompact abelian group of uncountable weight admits both a proper dense pseudocompact subgroup and a strictly larger pseudocompact group topology.This presentation will describe both the necessary new details of the argument and the historical development (useful tools, special cases). Among those who contributed essentially are: K.

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“…The articles [3] (3.5, 5.9), [4] (Problems 2.2, 3.2), and [27] (Problem 9) provide some references and discussion.] Question 1.…”

Section: ) a Dense Subgroup Of A Pseudocompact Group Is Itself Pseudomentioning

confidence: 99%

Pseudocompact groups: progress and problems

Comfort

2008

Topology and its Applications

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“…Our next example shows that this theorem does not generalize to solvable topological groups and thus resolves in negative Question 3.13 in [10] and Question 36 in [12]. Proof.…”

Section: Some Counterexamplesmentioning

confidence: 99%

“…This example (analyzed in details in Section 10) answers Question 3.13 in [10] and Question 36 in [12].…”

Section: Theoremmentioning

confidence: 99%

“…e: TS-Closed and h: TS-closed topological semigroups were introduced in 1969 by Stepp [7,8] who called them maximal and absolutely maximal semigroups, respectively. The study of h: TG-closed, p: TG-closed and c: TG-closed topological groups (called h-complete, hereditarily h-complete and c-compact topological groups, respectively) was initiated by Dikranjan and Tonolo [9] and continued by Dikranjan, Uspenskij [10], see the monograph of Lukàcs [11] and survey ( [12] §4) of Dikranjan and Shakhmatov. The study of e:pTG-closed paratopological groups was initiated by Banakh and Ravsky [13,14], who called them H-closed paratopological groups.…”

Section: Problemmentioning

confidence: 99%

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Categorically Closed Topological Groups

Банах

2017

Axioms

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Let C be a category whose objects are semigroups with topology and morphisms are closed semigroup relations, in particular, continuous homomorphisms. An object X of the category C is called C-closed if for each morphism Φ ⊂ X × Y in the category C the image Φ(X) = {y ∈ Y : ∃x ∈ X (x, y) ∈ Φ} is closed in Y. In the paper we survey existing and new results on topological groups, which are C-closed for various categories C of topologized semigroups.

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“…Markov did not explicitly introduce this topology, although he implicitly considered it in the form of algebraic closures of sets [20,19]. The Zariski topology was explicitly introduced only in 1977 by Bryant [5] under the name verbal topology; the term was introduced by Dikranjan and Shakhmatov [6]. This topology was studied in, e.g., [5,25,8,10,11].…”

Section: Problem 12 Find Necessary and Sufficient Conditions For Thmentioning

confidence: 99%

Topologizations of a set endowed with an action of a monoid

Банах

Протасов

Sipacheva

2014

Topology and its Applications

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Abstract. Given a set X and a family G of self-maps of X, we study the problem of the existence of a nondiscrete Hausdorff topology on X with respect to which all functions f ∈ G are continuous. A topology on X with this property is called a G-topology. The answer is given in terms of the Zariski G-topology ζ G on X, that is, the topology generated by the subbase consisting of the sets {x ∈ X : f (x) = g(x)} and {x ∈ X : f (x) = c}, where f, g ∈ G and c ∈ X. We prove that, for a countable monoid G ⊂ X X , X admits a non-discrete Hausdorff G-topology if and only if the Zariski G-topology ζ G is non-discrete; moreover, in this case, X admits 2 c hereditarily normal G-topologies.

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Selected topics from the structure theory of topological groups

Cited by 25 publications

References 36 publications

Pseudocompact groups: progress and problems

Pseudocompact groups: progress and problems

Categorically Closed Topological Groups

Topologizations of a set endowed with an action of a monoid

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