scholarly journals Filter games and pathological subgroups of a countable product of lines

2006 ◽  
Vol 81 (3) ◽  
pp. 321-350 ◽  
Author(s):  
Taras Banakh ◽  
Peter Nickolas ◽  
Manuel Sanchis
Keyword(s):  
Topological Properties ◽  
Topological Groups ◽  
Martin's Axiom ◽  
Martin’S Axiom ◽  
Countable Product ◽  
Free Ultrafilter

AbstractTo 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.

scholarly journals Martin's axiom and a regular topological space with uncountable net weight whose countable product is hereditarily separable and hereditarily Lindelöf

1987 ◽  
Vol 52 (2) ◽  
pp. 396-399
Author(s):  
Krzysztof Ciesielski
Keyword(s):  
Topological Space ◽  
Hausdorff Space ◽  
Continuum Hypothesis ◽  
Negative Answer ◽  
Martin's Axiom ◽  
Martin’S Axiom ◽  
Countable Product ◽  
Generic Filter ◽  
Cohen Forcing ◽  
The Continuum

In [1, p. 51] A. V. Arhangel'skiĭ, in connection with the problems of L-spaces and S-spaces, examined further the notions of hereditary separability and hereditary Lindelöfness. In particular he considered the following property P: “Every regular topological space has a countable net weight provided its countable product is hereditarily Lindelöf and hereditarily separable.” He noticed that the continuum hypothesis implies negation of the property P and posed a question: “Do Martin's Axiom and the negation of the continuum hypothesis imply P?” The purpose of this paper is to give a negative answer to this question.The set-theoretical and topological notation that we use is standard and can be found in [6] and [5] respectively.Throughout the paper we will use the notation H(X, Y) to denote the set of all finite functions from a set X to Y.Theorem. Con(ZFC) → Con(ZFC + MA + ¬CH + there exists a 0-dimensional Hausdorff space X such that nw(X) = с and nw(Y) = ω for any Y ϵ [X]<с).Proof. Let M be a model of ZFC satisfying CH and let F be an M-generic filter over the Cohen forcing {H(ω2 × ω2, 2), ⊃). Then f = ⋃F is a function and f: ω2 × ω2 → 2.

scholarly journals Construction of dual modules using Martin's Axiom

2008 ◽  
Vol 320 (6) ◽  
pp. 2388-2404
Author(s):  
Rüdiger Göbel ◽  
Sebastian Pokutta
Keyword(s):  
Martin's Axiom ◽  
Martin’S Axiom

scholarly journals Strongly Summable Ultrafilters, Union Ultrafilters, and the Trivial Sums Property

2016 ◽  
Vol 68 (1) ◽  
pp. 44-66 ◽  
Author(s):  
David J. Fernández Bretón
Keyword(s):  
Abelian Group ◽  
Partial Orders ◽  
Martin's Axiom ◽  
Martin’S Axiom ◽  
The Given

AbstractWe answer two questions of Hindman, Steprāns, and Strauss; namely, we prove that every strongly summable ultrafilter on an abelian group is sparse and has the trivial sums property. Moreover, we show that in most cases the sparseness of the given ultrafilter is a consequence of its being isomorphic to a union ultrafilter. However, this does not happen in all cases; we also construct (assuming Martin's Axiom for countable partial orders, i.e., , a strongly summable ultrafilter on the Boolean group that is not additively isomorphic to any union ultrafilter.

The equivalence of a generalized Martin's axiom to a combinatorial principle

1981 ◽  
Vol 46 (4) ◽  
pp. 817-821 ◽  
Author(s):  
William Weiss
Keyword(s):  
Combinatorial Principle ◽  
Martin's Axiom ◽  
Martin’S Axiom

AbstractA generalized version of Martin's axiom, called BACH, is shown to be equivalent to one of its combinatorial consequences, a generalization of P(c).

Countable dense homogeneous spaces under Martin’s axiom

1989 ◽  
Vol 65 (2) ◽  
pp. 153-164 ◽  
Author(s):  
Stewart Baldwin ◽  
Robert E. Beaudoin
Keyword(s):  
Homogeneous Spaces ◽  
Martin's Axiom ◽  
Martin’S Axiom
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