Analytic gaps
Stevo Todorčević (1996)
Fundamenta Mathematicae
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We investigate when two orthogonal families of sets of integers can be separated if one of them is analytic.
Stevo Todorčević (1996)
Fundamenta Mathematicae
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We investigate when two orthogonal families of sets of integers can be separated if one of them is analytic.
Carlos González (1995)
Fundamenta Mathematicae
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We investigate the relative consistency and independence of statements which imply the existence of various kinds of dense orders, including dense linear orders. We study as well the relationship between these statements and others involving partition properties. Since we work in ZF (i.e. without the Axiom of Choice), we also analyze the role that some weaker forms of AC play in this context
Alan Dow, Klaas Hart (2000)
Fundamenta Mathematicae
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The Open Colouring Axiom implies that the measure algebra cannot be embedded into P(ℕ)/fin. We also discuss errors in previous results on the embeddability of the measure algebra.
Stevo Todorčević (1998)
Fundamenta Mathematicae
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We prove that the quotient algebra P(ℕ)/I over any analytic ideal I on ℕ contains a Hausdorff gap.
Carlos Uzcátegui (1996)
Fundamenta Mathematicae
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We prove that the σ-ideal I(E) (of closed smooth sets with respect to a non-smooth Borel equivalence relation E) does not have the covering property. In fact, the same holds for any σ-ideal containing the closed transversals with respect to an equivalence relation generated by a countable group of homeomorphisms. As a consequence we show that I(E) does not have a Borel basis.
Umberto Zannier (1995)
Acta Arithmetica
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Chris Miller, Patrick Speissegger (1999)
Fundamenta Mathematicae
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The open core of a structure ℜ := (ℝ,<,...) is defined to be the reduct (in the sense of definability) of ℜ generated by all of its definable open sets. If the open core of ℜ is o-minimal, then the topological closure of any definable set has finitely many connected components. We show that if every definable subset of ℝ is finite or uncountable, or if ℜ defines addition and multiplication and every definable open subset of ℝ has finitely many connected components, then the open core...
Uri Abraham, Stevo Todorčević (1997)
Fundamenta Mathematicae
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A combinatorial statement concerning ideals of countable subsets of ω is introduced and proved to be consistent with the Continuum Hypothesis. This statement implies the Suslin Hypothesis, that all (ω, ω*)-gaps are Hausdorff, and that every coherent sequence on ω either almost includes or is orthogonal to some uncountable subset of ω.