Topologically invariant σ-ideals on Euclidean spaces

Банах, Тарас; Morayne, Michał; Rałowski, Robert; Żeberski, Szymon

doi:10.4064/fm231-2-1

Cited by 2 publications

(4 citation statements)

References 9 publications

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“…The investigation of (topologically) invariant σ-ideals with Borel or analytic base on some model topological spaces was initiated by the authors in [3] and [4]. In this paper we shall study and classify invariant σ-ideals with analytic base on (good) Cantor measure spaces.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Classifying invariant $\sigma $-ideals with analytic base on good Cantor measure spaces

Банах

Rałowski

Żeberski

2015

Proc. Amer. Math. Soc.

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Abstract. Let X be a zero-dimensional compact metrizable space endowed with a strictly positive continuous Borel σ-additive measure µ which is good in the sense that for any clopen subsets U, V ⊂ X with µ(U ) < µ(V ) there is a clopen set W ⊂ V with µ(W ) = µ(U ). We study σ-ideals with Borel base on X which are invariant under the action of the group Hµ(X) of measure-preserving homeomorphisms of (X, µ), and show that any such σ-ideal I is equal to one of seven σ-ideals:Here [X] ≤κ is the ideal consisting of subsets of cardiality ≤ κ in X, M is the ideal of meager subsets of X, N = {A ⊂ X : µ(A) = 0} is the ideal of null subsets of (X, µ), and E is the σ-ideal generated by closed null subsets of (X, µ).

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Classifying invariant $\sigma $-ideals with analytic base on good Cantor measure spaces

Банах

Rałowski

Żeberski

2015

Proc. Amer. Math. Soc.

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“…Topologically invariant ideals in the finite dimensional case were considered in paper [3] devoted to studying topologically invariant σ-ideals with Borel base on Euclidean spaces R n . To present the principal results, we need to recall some definitions.…”

Section: Introduction and Survey Of Principal Resultsmentioning

confidence: 99%

“…In [3] we proved that the ideal M of meager subsets of a Euclidean space R n is the largest topologically invariant σ-ideal with BP-base on R n . This is not true anymore for the Hilbert cube I ω as shown by the σ-ideal σD 0 of countable-dimensional subsets of I ω .…”

Section: Introduction and Survey Of Principal Resultsmentioning

confidence: 99%

“…In [3] we proved that the family of all non-trivial topologically invariant σ-ideals with analytic base on a Euclidean space R n contains the smallest element, namely the σ-ideal σC 0 generated by so called tame Cantor sets in R n . A similar fact holds also for topologically invariant ideals with an analytic base on the Hilbert cube I ω .…”

Section: Introduction and Survey Of Principal Resultsmentioning

confidence: 99%

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Topologically invariant σ-ideals on the Hilbert cube

et al. 2015

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We study and classify topologically invariant σ-ideals with Borel base on the Hilbert cube and evaluate their cardinal characteristics. One of the results of this paper solves (positively) a known problem whether the minimal cardinalities of the families of Cantor sets covering the unit interval and the Hilbert cube are the same.1991 Mathematics Subject Classification. 03E15; 03E17; 54H05; 55M10; 57N20.

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Topologically invariant σ-ideals on Euclidean spaces

Abstract: We study and classify topologically invariant σ-ideals with an analytic base on Euclidean spaces and evaluate the cardinal characteristics of such ideals.

Cited by 2 publications

References 9 publications

Classifying invariant $\sigma $-ideals with analytic base on good Cantor measure spaces

Classifying invariant $\sigma $-ideals with analytic base on good Cantor measure spaces

Topologically invariant σ-ideals on the Hilbert cube

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