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Filter games and pathological subgroups of a countable product of lines

Part of: Game theory Generalities Set theory Topological and differentiable algebraic systems Connections with other structures, applications Spaces with richer structures Fairly general properties Peculiar spaces

Published online by Cambridge University Press:  09 April 2009

Taras Banakh ,
Peter Nickolas  and
Manuel Sanchis
Taras Banakh
Affiliation:
Instytut MatematykiAkademia Świetokrzyska Świetokrzyska 15 KielcePoland and Department of Mathematics Ivan Franko Lviv National UniversityUniversytetska 1 Lviv 79000Ukraina e-mail: tbanakh@franko.lviv.ua
Peter Nickolas
Affiliation:
School of Mathematics and Applied Statistics University of WollongongWollongong NSW 2522Australia e-mail: peter@uow.edu.au
Manuel Sanchis
Affiliation:
Departament de Matemàtiques Universitat Jaume ICampus de Penyeta Roja s/n 12071 CastellónSpain e-mail: sanchis@mat.uji.es
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Abstract

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To each filter ℱ on ω, a certain linear subalgebra A(ℱ) of Rω, the countable product of lines, is assigned. This algebra is shown to have many interesting topological properties, depending on the properties of the filter ℱ. For example, if ℱ is a free ultrafilter, then A(ℱ) is a Baire subalgebra of ℱω for which the game OF introduced by Tkachenko is undetermined (this resolves a problem of Hernández, Robbie and Tkachenko); and if ℱ1 and ℱ2 are two free filters on ω that are not near coherent (such filters exist under Martin's Axiom), then A (ℱ1) and A(ℱ2) are two o-bounded and OF-undetermined subalgebras of ℱω whose product A(ℱ1) × A(ℱ2) 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 ℱω 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.

Keywords

open-finite game, o-bounded group, filter game, near coherence of filters.

MSC classification

Secondary: 03E35: Consistency and independence results 03E50: Continuum hypothesis and Martin's axiom 03E60: Determinacy principles 22A05: Structure of general topological groups 54A35: Consistency and independence results 54D80: Special constructions of spaces (spaces of ultrafilters, etc.) 54E52: Baire category, Baire spaces 54G15: Pathological spaces 54H11: Topological groups 54H12: Topological lattices, etc. 54H13: Topological fields, rings, etc. 91A44: Games involving topology or set theory
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 81 , Issue 3 , December 2006 , pp. 321 - 350
Copyright
Copyright © Australian Mathematical Society 2006

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