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

Tsaban, Boaz

doi:10.1090/s0002-9939-05-08034-2

Cited by 22 publications

(23 citation statements)

References 16 publications

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“…In fact, the spaces L\, L 2 have the Menger property, which is stronger than o-boundedness. Another result of this sort can be also found in [25].…”

Section: Introductionsupporting

confidence: 57%

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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“…In fact, the spaces L\, L 2 have the Menger property, which is stronger than o-boundedness. Another result of this sort can be also found in [25].…”

Section: Introductionsupporting

confidence: 57%

Filter games and pathological subgroups of a countable product of lines

Банах¹,

Nickolas²,

Sanchis³

2006

J. Aust. Math. Soc.

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“…Since the γ -property is linearly σ -additive, hereditary for closed subsets, and preserved by continuous images, there is a subgroup of reals that satisfies the γ -property [29]. For the reader's convenience we reproduce the subgroup in [37].…”

Section: Theorem 31 ( [35]) the Axiom Of Coanalytic Determinacy Implies That Every Menger Coanalytic Topological Group Is σ -Compactmentioning

confidence: 99%

“…By using a similar argument as in Theorem 3.2, we can obtain a coanalytic γ -subgroup of reals denoted by G T . Notice that T is a closed subset of G T (see [37]) and not productively Lindelöf . Every closed subset of a productively Lindelöf space is productively Lindelöf.…”

Section: Claim G H Is Coanalyticmentioning

confidence: 99%

A coanalytic Menger group that is not $\sigma$-compact

Tokgöz¹

2018

Turk J Math

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“…Next we examine products in the class of Gerlits-Nagy spaces. Corollary 47 solves Problem 6.6 of [55] and Problems 3.1 through 3.3 of Samet and Tsaban, listed in Section 3 of [54]:…”

Section: Gerlits-nagy Spaces and Productsmentioning

confidence: 99%

Lindelöf indestructibility, topological games and selection principles

Scheepers

Tall

2010

Fund. Math.

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Arhangel'skii proved that if a first countable Hausdorff space is Lindelöf, then its cardinality is at most 2 ℵ 0 . Such a clean upper bound for Lindelöf spaces in the larger class of spaces whose points are G δ has been more elusive. In this paper we continue the agenda started in [50], of considering the cardinality problem for spaces satisfying stronger versions of the Lindelöf property. Infinite games and selection principles, especially the Rothberger property, are essential tools in our investigations 1 .A topological space is Lindelöf if each open cover contains a countable subset that covers the space. Alexandrov asked if in the class of first countable Hausdorff spaces, every Lindelöf space has cardinality at most 2 ℵ0 . Arhangel'skii [1] proved that the answer is "yes". This focuses attention on the larger class of spaces in which "points are G δ " -i.e., the class of topological spaces in which each point is an intersection of countably many open sets. Such spaces are T 1 but not necessarily Hausdorff. Arhangel'skii showed that in the class of spaces with points G δ each Lindelöf space has cardinality less than the least measurable cardinal. Juhász [26] showed that this bound is sharp: There are such Lindelöf spaces of arbitrary large cardinality below the least measurable cardinal. Juhász's examples are not Hausdorff spaces and the cardinality of the underlying spaces has countable cofinality. Shelah [44] showed that no Lindelöf space with points G δ can be of weakly compact cardinality. Gorelic [19] ℵ0 is (modulo large cardinals) independent. This independence can also be obtained in another way if the separation hypothesis is weakened: A. Dow [13] showed that adding ℵ 1 Cohen reals converts every ground model Lindelöf space to an indestructibly Lindelöf space in the generic extension. This gives the consistency of the existence of non-T 2 indestructibly Lindelöf spaces with points G δ of arbitrary large cardinality below the first measurable cardinal: Juhász's spaces from the ground model retain their cardinality but acquire Lindelöf indestructibility, and measurable cardinals from the ground model remain measurable.[50] Theorem 3 gives a combinatorial characterization of indestructibly Lindelöf. In Section 1, starting from this combinatorial characterization, we show how to characterize indestructibly Lindelöf game-theoretically (Theorem 1) and examine the determinacy of this game. This analysis leads us to two natural strengthenings of indestructibly Lindelöf. For one of these strengthenings, all spaces with points G δ and this strengthening have cardinality ≤ 2 ℵ0 (Theorem 2). The classical selection property introduced by Rothberger is the other natural strengthening (Corollary 10). These topics are illustrated with some examples collected in Section 4.In Section 2 we focus on the Rothberger property. In [15], using the technique of n-dowments, it was shown that a variety of non-covering and non-generalizedmetric properties were preserved by Cohen reals. Since these arguments mimic mea...

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

Cited by 22 publications

References 16 publications

Filter games and pathological subgroups of a countable product of lines

Filter games and pathological subgroups of a countable product of lines

A coanalytic Menger group that is not $\sigma$-compact

Lindelöf indestructibility, topological games and selection principles

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