Kenilworth

Mátrai, Tamás

doi:10.1090/s0002-9939-08-09615-9

Cited by 5 publications

(12 citation statements)

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“…It can be shown that for Mátrai's G δ ideal J from [14] the following is true: if J = n V n with V n ⊆ K(E) open and downward closed, then add I (V n ) = 1 for some n. In this respect J is similar to the ideal I 1 from the theorem above. These two ideals are however distinct.…”

Section: Unboundedness Of the Rankmentioning

confidence: 90%

“…(There exist overlaps among the classes of ideals listed below, which I have not investigated.) On the other hand, as follows from [14], there do exist G δ ideals of compact subsets of 2 ω that do not have ( * ).…”

Section: Basic Facts and Examplesmentioning

confidence: 99%

“…Recently, a new phenomenon among G δ ideals of compact sets was discovered by Mátrai in [14]. He constructed a G δ ideal of compact subsets of 2 ω containing all singletons and such that each dense G δ subset of 2 ω contains a compact set not in the ideal.…”

Section: Introductionmentioning

confidence: 99%

“…At this point, this property seems somewhat pathological for G δ ideals, and condition ( * ) delineates a natural class of G δ ideals avoiding it. (Sections 2, 3, and 4 of the present paper were, however, mostly completed before the appearance of [14].) The aim of presenting the examples in Section 2 is to show that natural G δ ideals do have property ( * ).…”

Section: Introductionmentioning

confidence: 99%

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$G_\delta$ ideals of compact sets

Solecki¹

2011

J. Eur. Math. Soc.

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Abstract. We investigate the structure of G δ ideals of compact sets. We define a class of G δ ideals of compact sets that, on the one hand, avoids certain phenomena present among general G δ ideals of compact sets and, on the other hand, includes all naturally occurring G δ ideals of compact sets. We prove structural theorems for ideals in this class, and we describe how this class is placed among all G δ ideals. In particular, we establish a result representing ideals in this class via the meager ideal. This result is analogous to Choquet's theorem representing alternating capacities of order ∞ via Borel probability measures. Methods coming from the structure theory of Banach spaces are used in constructing important examples of G δ ideals outside of our class.

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Section: Unboundedness Of the Rankmentioning

confidence: 90%

Section: Basic Facts and Examplesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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$G_\delta$ ideals of compact sets

Solecki¹

2011

J. Eur. Math. Soc.

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“…We study the place F σ ideals occupy in the Tukey order. As a point of reference we use the following, now standard, diagram which summarizes the known Tukey reductions among the well studied directed sets; see [3,5,8,10,12,15,17,18,19]. As usual, in the diagram, an arrow denotes reduction and the absence of arrows non-reduction.…”

Section: Introductionmentioning

confidence: 99%

Tukey order among $F_σ$ ideals

He¹,

Hrušák²,

Rojas-Rebolledo³

et al. 2020

Preprint

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We investigate the Tukey order in the class of Fσ ideals of subsets of ω. We show that no nontrivial Fσ ideal is Tukey below a G δ σ-ideal of compact sets. We introduce the notions of flat ideals and gradually flat ideals. We prove a dichotomy theorem for flat ideals isolating gradual flatness as the side of the dichotomy that is structurally good. We give diverse characterizations of gradual flatness among flat ideals using Tukey reductions and games. For example, we show that all gradually flat ideals are Tukey below the density zero ideal Z 0 .

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On a σ-ideal of compact sets

Mátrai

2010

Topology and its Applications

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Kenilworth

Cited by 5 publications

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$G_\delta$ ideals of compact sets

$G_\delta$ ideals of compact sets

Tukey order among $F_σ$ ideals

On a σ-ideal of compact sets

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