Filter games and pathological subgroups of a countable product of lines

Authors

Taras Banakh, Lviv State Ivan Franko University
Peter Nickolas, University of WollongongFollow
Manuel Sanchis, Universitat Jaume I de Castellon

RIS ID

17876

Publication Details

Banakh, T., Nickolas, P. R. & Sanchis, M. (2006). Filter games and pathological subgroups of a countable product of lines. Journal of the Australian Mathematical Society, 81 (3), 321-350.

Abstract

To each filter F on ω, a certain linear subalgebra A(F) of R^ω, 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 R^ω 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 R^ω 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 R^ω 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.