Questions on generalised Baire spaces

Khomskii, Yurii; Laguzzi, Giorgio; Löwe, Benedikt; Sharankou, Ilya

doi:10.1002/malq.201600051

Cited by 23 publications

(26 citation statements)

References 50 publications

(105 reference statements)

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“…In case κ is strongly inaccessible in the ground model -and weakly inaccessible in the extension -the consistency of cov(M κ ) < d κ ( = * ) answers a question of Matet and Shelah [MS,Section 4] (see also [KLLS,Question 3.8,part 2]).…”

Section: Generalized Hechler Forcingmentioning

confidence: 99%

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Cichoń’s diagram for uncountable cardinals

et al. 2018

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We develop a version of Cichoń's diagram for cardinal invariants on the generalized Cantor space 2 κ or the generalized Baire space κ κ where κ is an uncountable regular cardinal. For strongly inaccessible κ, many of the ZFC-results about the order relationship of the cardinal invariants which hold for ω generalize; for example we obtain a natural generalization of the Bartoszyński-Raisonnier-Stern Theorem. We also prove a number of independence results, both with < κ-support iterations and κ-support iterations and products, showing that we consistently have strict inequality between some of the cardinal invariants.

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Section: Generalized Hechler Forcingmentioning

confidence: 99%

“…Question 22. (Matet,Shelah [MS,Section 4], see also [KLLS,Question 3.8, part 1]) Is it consistent that κ is a successor cardinal and d κ ( = * ) < d κ ?…”

Section: (Landvermentioning

confidence: 99%

Cichoń’s diagram for uncountable cardinals

et al. 2018

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“…This paper is concerned with forcings consisting of uncountable trees. In particular we focus on some issues about pure decision and Cohen κ-reals, attacking some points raised in [11,Question 3.20]. The main part of the paper is centered around the investigation of amoeba forcings, and more generally, the effects of adding uncountable generic trees over models of ZFC.…”

Section: Introductionmentioning

confidence: 99%

“…We remark that this reaserch field has largely spread out in the last years among set theorists. In particular, the problems analysed in this paper are part of those collected in [11], which is the output of a series of workshops which took place in Amsterdam (November 2014), Hamburg (September 2015) and Bonn (September 2016).…”

Section: Introductionmentioning

confidence: 99%

Uncountable Trees and Cohen -Reals

Laguzzi¹

2019

J. symb. log.

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We investigate some versions of amoeba for tree-forcings in the generalized Cantor and Baire spaces. This answers [10, Question 3.20] and generalizes a line of research that in the standard case has been studied in [11], [13], and [7]. Moreover, we also answer questions posed in [3] by Friedman, Khomskii, and Kulikov, about the relationships between regularity properties at uncountable cardinals. We show ${\bf{\Sigma }}_1^1$-counterexamples to some regularity properties related to trees without club splitting. In particular we prove a strong relationship between the Ramsey and the Baire properties, in slight contrast with the standard case.

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“…In particular, we would like to draw attention both on the sometimes surprising results emerging in this area, and on the remarkably wide variety of open problems and lines for further research that are consequently revealed. Recent work, by various people, on generalized Baire spaces and related set-theoretic issues (see, for instance, Friedman, Khomskii and Kulikov [15] and Khomskii, Laguzzi, Löwe and Sharankou [23]) should be thought as complementary in this direction, possibly with important underlying connections waiting to surface.…”

Section: Introductionmentioning

confidence: 99%

Long reals

Asperó¹,

Tsaprounis²

2018

JLA

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The familiar continuum R of real numbers is obtained by a well-known procedure which, starting with the set of natural numbers N = ω , produces in a canonical fashion the field of rationals Q and, then, the field R as the completion of Q under Cauchy sequences (or, equivalently, using Dedekind cuts). In this article, we replace ω by any infinite suitably closed ordinal κ in the above construction and, using the natural (Hessenberg) ordinal operations, we obtain the corresponding field κ-R, which we call the field of the κ-reals. Subsequently, we study the properties of the various fields κ-R and develop their general theory, mainly from the set-theoretic perspective. For example, we investigate their connection with standard themes such as forcing and descriptive set theory.2010 Mathematics Subject Classification 12L15, 12J15, 12L99 (primary); 03E10, 03E15 (secondary)

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Questions on generalised Baire spaces

Abstract: We provide a list of open problems in the research area of generalised Baire spaces, compiled with the help of the participants of two workshops held in Amsterdam (2014) and Hamburg (2015).

Cited by 23 publications

References 50 publications

Cichoń’s diagram for uncountable cardinals

Cichoń’s diagram for uncountable cardinals

Uncountable Trees and Cohen -Reals

Long reals

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