Filter games and pathological subgroups of a countable product of lines

Abstract

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.

The combinatorics of the Baer-Specker group

“…and results similar to these, were previously obtained in [4,25,5] and possibly in additional places.…”

Selection principles and special sets of reals

“…is not provably preserved under cartesian products [5,34,65]. If G is analytic and does not satisfy S 1 (Ω nbd , Γ), then G 2 does not satisfy S fin (O nbd , O).…”

Squares of Menger-bounded groups

“…Question 1 of Banakh, Nickolas, and Sanchis [5] asks whether each Mengerbounded subgroup of C N (with coordinate-wise addition) is mixable or o F -bounded for some filter F. As it is proved there that mixable Menger-bounded groups are Scheepers bounded, and the same holds for groups which are o F -bounded for some filter F, we obtain a negative answer to both questions: The groups we construct are, in particular, subgroups of C N . The problem of whether, consistently, every Menger-bounded group is Scheepersbounded is yet to be addressed.…”