A parallel to the null ideal for inaccessible $$\lambda $$ λ : Part I

“…• In Section 1 we repeat some key definitions and results from [She17], introduce some notations and finally define the notion of a strengthened Galois-Tukey connection. • In Section 2 we prove preservation theorems for iterations of <κ-and κ-support.…”

Section: Overview Of the Papermentioning

confidence: 99%

“…We will be interested in a natural generalization of such properties to the spaces 2 κ and κ κ for uncountable (and in this paper: always inaccessible) cardinals κ. This area of research has progressed quickly in recent years; [KL + 16] collected many questions inspired by workshops on generalized reals, and several recent results can be found in [BB + 18], [FL17], [She17], [CS19].…”

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confidence: 99%

“…In [She17] the third author introduced a generalization ( 1 ) Q κ of the random forcing to 2 κ for inaccessible κ. The forcing Q κ is κ-strategically closed, satisfies the κ + -chain condition and for weakly compact κ is κ κbounding.…”

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confidence: 99%

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The higher Cichoń diagram

2021

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For a strongly inacessible cardinal κ, we investigate the relationships between the following ideals: (1) the ideal of meager sets in the <κ-box product topology, (2) the ideal of "null" sets in the sense of [She17], (3) the ideal of nowhere stationary subsets of a (naturally defined) stationary set S κ pr ⊆ κ. In particular, we analyze the provable inequalities between the cardinal characteristics for these ideals, and we give consistency results showing that certain inequalities are unprovable. While some results from the classical case (κ = ω) can be easily generalized to our setting, some key results (such as a Fubini property for the ideal of null sets) do not hold; this leads to the surprising inequality cov(null) ≤ non(null). Also, concepts that did not exist in the classical case (in particular, the notion of stationary sets) will turn out to be relevant. We construct several models to distinguish the various cardinal characteristics; the main tools are iterations with <κ-support (and a strong "Knaster" version of κ +-c.c.) and one iteration with ≤κ-support (and a version of κ-properness).

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Special subsets of the generalized Cantor space and generalized Baire space

Korch

Weiss

2020

Mathematical Logic Qtrly

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In this paper, we are interested in parallels to the classical notions of special subsets in R defined in the generalized Cantor and Baire spaces (2 κ and κ κ). We consider generalizations of the well-known classes of special subsets, like Lusin sets, strongly null sets, concentrated sets, perfectly meagre sets, σ-sets, γ-sets, sets with the Menger, the Rothberger, or the Hurewicz property, but also of some less-know classes like X-small sets, meagre additive sets, Ramsey null sets, Marczewski, Silver, Miller, and Laver-null sets. We notice that many classical theorems regarding these classes can be relatively easy generalized to higher cardinals although sometimes with some additional assumptions. This paper serves as a catalogue of such results along with some other generalizations and open problems.

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A parallel to the null ideal for inaccessible $$\lambda $$ λ : Part I

Cited by 13 publications

References 17 publications

Questions on generalised Baire spaces

Questions on generalised Baire spaces

The higher Cichoń diagram

Special subsets of the generalized Cantor space and generalized Baire space

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