Mathias Forcing and Combinatorial Covering Properties of Filters

Chodounský, David; Repovš, Dušan; Zdomskyy, Lyubomyr

doi:10.1017/jsl.2014.73

Cited by 28 publications

(32 citation statements)

References 23 publications

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“…According to [5,Theorem 3.5], Menger filters are precisely those called Canjar filters. Also, by [6,Proposition 2], d = c implies there is a Canjar (thus, Menger) ultrafilter.…”

Section: Introductionmentioning

confidence: 99%

A non-discrete space X with Cp(X) Menger at infinity

Bella

Hernández-Gutiérrez

2019

Appl. Gen. Topol.

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“…According to [5,Theorem 3.5], Menger filters are precisely those called Canjar filters. Also, by [6,Proposition 2], d = c implies there is a Canjar (thus, Menger) ultrafilter.…”

Section: Introductionmentioning

confidence: 99%

A non-discrete space X with Cp(X) Menger at infinity

Bella

Hernández-Gutiérrez

2019

Appl. Gen. Topol.

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“…The Mathias-Prikry and the Laver type forcings were introduced in [Mat77] and [Gro87] respectively. Recently, properties of these forcings were characterized in terms of properties of associated filters, see [BH09,ChRZ14,GHMC14,HM14]. We continue this line of research, and investigate forcings associated with coideals.…”

Section: Introductionmentioning

confidence: 90%

“…It is easy to see that r is an (X ) generic real iff G r (X ) is a generic filter on (X ). Properties of (F) when F is an ultrafilter were studied in [Can88] and for F a general filter in [HM14,ChRZ14]. Since (F) is σ-centered, it always adds an unbounded real.…”

Section: Preliminariesmentioning

confidence: 99%

Mathias–Prikry and Laver type forcing; summable ideals, coideals, and +-selective filters

2016

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We study the Mathias-Prikry and the Laver type forcings associated with filters and coideals. We isolate a crucial combinatorial property of Mathias reals, and prove that Mathias-Prikry forcings with summable ideals are all mutually bi-embeddable. We show that Mathias forcing associated with the complement of an analytic ideal always adds a dominating real. We also characterize filters for which the associated Mathias-Prikry forcing does not add eventually different reals, and show that they are countably generated provided they are Borel. We give a characterization of ω-hitting and ω-splitting families which retain their property in the extension by a Laver type forcing associated with a coideal.

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“…The proof is identical to that of Theorem 4.6, replacing Λ(X) by B and using Theorem 5.5 instead of Proposition 4. 5.…”

Section: Richer Coversmentioning

confidence: 99%

Algebra, selections, and additive Ramsey theory

Tsaban

2018

Fund. Math.

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Abstract. Hindman's celebrated Finite Sums Theorem, and its high-dimensional version due to Milliken and Taylor, are extended from covers of countable sets to covers of arbitrary topological spaces with Menger's classic covering property. The methods include, in addition to Hurewicz's game theoretic characterization of Menger's property, extensions of the classic idempotent theory in the Stone-Čech compactification of semigroups, and of the more recent theory of selection principles. This provides strong versions of the mentioned celebrated theorems, where the monochromatic substructures are large, beyond infinitude, in an analytic sense. Reducing the main theorems to the purely combinatorial setting, we obtain nontrivial consequences concerning uncountable cardinal characteristics of the continuum.The main results, modulo technical refinements, are of the following type (definitions provided in the main text): Let X be a Menger space, and U be an infinite open cover of X. Consider the complete graph, whose vertices are the open sets in X. For each finite coloring of the vertices and edges of this graph, there are disjoint finite subsets F 1 , F 2 , . . . of the cover U whose unions V 1 := F 1 , V 2 := F 2 , . . . have the following properties:(1) The sets n∈F V n and n∈H V n are distinct for all nonempty finite sets F < H.(2) All vertices n∈F V n , for nonempty finite sets F , are of the same color.(3) All edges n∈F V n , n∈H V n , for nonempty finite sets F < H, have the same color. (4) The family {V 1 , V 2 , . . . } is an open cover of X. A self-contained introduction to the necessary parts of the needed theories is provided.

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Mathias Forcing and Combinatorial Covering Properties of Filters

Abstract: We give topological characterizations of filters F on such that the Mathias forcing MF adds no dominating reals or preserves ground model unbounded families. This allows us to answer some questions of Brendle, Guzmán, Hrušák, Martínez, Minami, and Tsaban.

Cited by 28 publications

References 23 publications

A non-discrete space X with Cp(X) Menger at infinity

A non-discrete space X with Cp(X) Menger at infinity

Mathias–Prikry and Laver type forcing; summable ideals, coideals, and +-selective filters

Algebra, selections, and additive Ramsey theory

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