Topologically invariant σ-ideals on the Hilbert cube

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

doi:10.1007/s11856-015-1235-z

Cited by 3 publications

(6 citation statements)

References 15 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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“…There are other natural spaces where it would be interesting and useful to know these properties. The Hilbert cube case is examined in [2].…”

Section: Introductionmentioning

confidence: 99%

Topologically invariant σ-ideals on Euclidean spaces

et al. 2015

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“…K-Universal sets for various classes K often appear in topology. A classical example of such set is the Sierpiński Carpet M 2 1 , known to be a K-universal set for the family K of all (closed) nowhere dense subsets of the square I 2 = [0, 1] 2 (see [14]). The Sierpiński Carpet M 2 1 is one of the Menger cubes M n k , which are K-universal for the family K of all k-dimensional compact subsets of the n-dimensional cube I n (see [15], [8, §4.1]).…”

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confidence: 99%

“…A classical example of such set is the Sierpiński Carpet M 2 1 , known to be a K-universal set for the family K of all (closed) nowhere dense subsets of the square I 2 = [0, 1] 2 (see [14]). The Sierpiński Carpet M 2 1 is one of the Menger cubes M n k , which are K-universal for the family K of all k-dimensional compact subsets of the n-dimensional cube I n (see [15], [8, §4.1]). An analogue of the Sierpiński Carpet exists also in the Hilbert cube I ω , which contains a Z 0 -universal set for the family Z 0 of closed nowhere dense subsets of I ω (see [3]).…”

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confidence: 99%