#### RIS ID

17876

#### 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 F_{1} and F_{2} are two free filters on ω that are not near coherent (such filters exist under Martin's Axiom), then *A* (F_{1}) and *A*(F_{2}) are two *o*-bounded and OF-undetermined subalgebras of R^{ω} whose product *A*(F_{1}) × *A*(F_{2}) 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.

## 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.