J. Aust. Math. Soc.
81 (2006), 321350

Filter games and pathological subgroups of a countable product of lines

Taras Banakh
Instytut Matematyki
Akademia Swietokrzyska
Swietokrzyska 15
Kielce
Poland
and
Department of Mathematics
Ivan Franko Lviv National University
Universytetska 1
Lviv 79000
Ukraina
tbanakh@franko.lviv.ua


Peter Nickolas
School of Mathematics and
Applied Statistics
University of Wollongong
Wollongong
NSW 2522
Australia
peter@uow.edu.au



Manuel Sanchis
Departament de Matemàtiques
Universitat Jaume I
Campus de Penyeta Roja
s/n 12071
Castellón
Spain
sanchis@mat.uji.es



Abstract

To each filter on , a certain linear subalgebra
of , 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
is a Baire subalgebra of
for which the game OF introduced by Tkachenko is
undetermined (this resolves a problem of
Hernández, Robbie and Tkachenko); and if
and
are two free filters on
that are not near coherent (such filters exist
under Martin's Axiom), then
and
are two
obounded and OFundetermined subalgebras of
whose product
is OFdetermined and not
obounded (this resolves a problem of Tkachenko).
It is also shown that the statement that the
product of two
obounded subrings of
is obounded is equivalent to the settheoretic
principle NCF (Near Coherence of Filters); this
suggests that Tkachenko's question on the
productivity of the class of
obounded topological groups may be undecidable in
ZFC.

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