Some properties of o-bounded and strictly o-bounded groups

Hernández, Constancio; Tkachenko, Mikhail; Robbie, Desmond A.

doi:10.4995/agt.2000.3022

Cited by 20 publications

(26 citation statements)

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“…-undetermined) if G has the respective property as a subset of G. If & is the collection of all the singletons of co, then we shall omit the subscript & and shall speak about the game OF and II-, I-, o-bounded, OF-determined and OF-undetermined sets in place of the game OF^ and 11^-, Ijr-, o& -bounded, OF^-determined and OF^-undetermined sets, respectively. It should be mentioned that in [1,10,11,22,23], II-bounded groups are called 'strictly o-bounded', but we prefer the term 'II-bounded', accepted also in [2].…”

Section: Andmentioning

confidence: 99%

“…A reflection of this is the fact that the first claimed example of a metrizable obounded non-II-bounded group H, presented by Hernandez in [10, Example 6.1] (and exploited in [22, page 195], [1,Theorem 4] and [11,Example 2.12]), turned out to be incorrect. (By [1], the group H is analytic, and being non-cr-bounded, is not o-bounded, according to Theorem 1 (3).)…”

Section: Andmentioning

confidence: 99%

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Filter games and pathological subgroups of a countable product of lines

Банах¹,

Nickolas²,

Sanchis³

2006

J. Aust. Math. Soc.

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To each filter & on co, a certain linear subalgebra A (^) of R™, the countable product of lines, is assigned. This algebra is shown to have many interesting topological properties, depending on the properties of the filter &. For example, if & is a free ultrafilter, then A(^) is a Baire subalgebra of R" for which the game OF introduced by Tkachenko is undetermined (this resolves a problem of Hernandez, Robbie and Tkachenko); and if & x and ^2 are two free filters on a> that are not near coherent (such filters exist under Martin's Axiom), then A(&{) and A(J? 2 ) are two o-bounded and OF-undetermined subalgebras of U." whose product A(^\) x A(^i) is OF-determined and not o-bounded (this resolves a problem of Tkachenko). It is also shown that the statement that the product of two o-bounded subrings of R" is o-bounded is equivalent to the set-theoretic principle NCF (Near Coherence of Filters); this suggests that Tkachenko's question on the productivity of the class of o-bounded topological groups may be undecidable in ZFC.

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Section: Andmentioning

confidence: 99%

Section: Andmentioning

confidence: 99%

Filter games and pathological subgroups of a countable product of lines

Банах¹,

Nickolas²,

Sanchis³

2006

J. Aust. Math. Soc.

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“…In [15,11], a unified framework for topological diagonalizations is established, which turns out to be closely related to several notions which appear in a recently flourishing study of topological groups in terms of their covering properties (see, e.g., [18,9,10,12,13] and references therein). A comprehensive study of these interrelations is currently being carried out by Babinkostova, Kočinac, and Scheepers [1].…”

Section: Introductionmentioning

confidence: 99%

𝑜-bounded groups and other topological groups with strong combinatorial properties

Tsaban

2005

Proc. Amer. Math. Soc.

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Abstract. We construct several topological groups with very strong combinatorial properties. In particular, we give simple examples of subgroups of R (thus strictly o-bounded) which have the Menger and Hurewicz properties but are not σ-compact, and show that the product of two o-bounded subgroups of R N may fail to be o-bounded, even when they satisfy the stronger property S 1 (B Ω , B Ω ). This solves a problem of Tkačenko and Hernandez, and extends independent solutions of Krawczyk and Michalewski and of Banakh, Nickolas, and Sanchis. We also construct separable metrizable groups G of size continuum such that every countable Borel ω-cover of G contains a γ-cover of G.

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“…We introduce the notion of relator Menger space and explore its properties in a similar way as it was done in topological groups (see [3], [8], [9]). Definition 2.1 (Ko£inac).…”

Section: Relator Menger and Related Spacesmentioning

confidence: 99%

“…In a sense, the properties and games which we consider are related to those from [3], [5], [8], [9], [11] and [12].…”

Section: Introductionmentioning

confidence: 99%