Universally meager sets, II

Zakrzewski, Piotr

doi:10.1016/j.topol.2008.05.005

Cited by 14 publications

(12 citation statements)

References 6 publications

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“…Let us cite the following result (see [24,Theorem 1.2]) which completely solves the Baire case and strongly improves our previous results on this topic:…”

Section: Measure and Categorysupporting

confidence: 76%

“…A set X in a topological space is 1. always of the first category (AF C) if it is meager relative to every perfect set; 2. universally meager (UM or AF C) if it does not contain a continuous and one-to-one image of a set of second category (see [13], [14], or [23]); 3. meager additive (M * ) if M + X is meager for every meager set M (see [4]). …”

Section: Definitionsmentioning

confidence: 99%

“…It was proved in [23] that a set X is universally meager if and only if each Borel isomorphic image of X in R is meager, and that a product of two universally meager sets is universally meager. Notice also that UM ⊆ AF C. Suppose that F is an arbitrary family of nonempty subsets of a set X .…”

Section: Definitionsmentioning

confidence: 99%

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Uniform Algebras in the Cantor and Baire Spaces

Nowik¹,

Reardon²

2008

Journal of Applied Analysis

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Abstract. For each pointclass Γ ⊆ P (2 ω ) define U [Γ] as the collection of all X ⊆ 2 ω such that the preimage f −1 (X) belongs to Γ for each continuous f : 2 ω → 2 ω . We study the properties of and possible relationships among the classes U [Γ], where Γ ranges over the σ-algebras (l), (m), the completely Ramsey sets, and the sets with the Baire property. We also prove some results about cardinal coefficients of U [Γ] for the general case of Marczewski-Burstin representable σ-algebras Γ. We finish by posing some unsolved problems.

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“…Let us cite the following result (see [24,Theorem 1.2]) which completely solves the Baire case and strongly improves our previous results on this topic:…”

Section: Measure and Categorysupporting

confidence: 76%

Section: Definitionsmentioning

confidence: 99%

See 1 more Smart Citation

Uniform Algebras in the Cantor and Baire Spaces

Nowik¹,

Reardon²

2008

Journal of Applied Analysis

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“…The notion of perfectly meager sets has some natural modifications; see [10], [14]. Definition 4.4 (Zakrzewski).…”

Section: Corollary 43 Let F Be a Family Of Subsets Of T Such That Dmentioning

confidence: 99%

Dirichlet sets, Erdős-Kunen-Mauldin theorem, and analytic subgroups of the reals

Eliáš

2010

Proc. Amer. Math. Soc.

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Abstract. We prove strengthenings of two well-known theorems related to the Lebesgue measure and additive structure of the real line. The first one is a theorem of Erdős, Kunen, and Mauldin stating that for every perfect set there exists a perfect set of measure zero such that their algebraic sum is the whole real line. The other is Laczkovich's theorem saying that every proper analytic subgroup of the real line is included in an F σ set of measure zero. Using the strengthened theorems we generalize the fact that permitted sets for families of trigonometric thin sets are perfectly meager.

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“…A subset X of R is universally meager [28] if every Borel isomorphic image of X is meager in R. A subset X of R is a λ-set [18] if every countable subset of X is a G δ -set in X. ≤ω iff X is a λ-set with the Hurewicz property [13,Theorem 3].…”

Section: Definition 43mentioning

confidence: 99%

Menger subsets of the Sorgenfrey line

Sakai

2009

Proc. Amer. Math. Soc.

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Abstract. A space X is said to have the Menger property if for every sequence {U n : n ∈ ω} of open covers of X, there are finite subfamilies V n ⊂ U n (n ∈ ω) such that n∈ω V n is a cover of X. Let i : S → R be the identity map from the Sorgenfrey line onto the real line and let X S = i −1 (X) for X ⊂ R. Lelek noted in 1964 that for every Lusin set L in R, L S has the Menger property. In this paper we further investigate Menger subsets of the Sorgenfrey line. Among other things, we show: (1) If X S has the Menger property, then X has Marczewski's property (s 0 ). (2) Let X be a zero-dimensional separable metric space. If X has a countable subset Q satisfying that X \ A has the Menger property for every countable set A ⊂ X \ Q, then there is an embedding e : X → R such that e(X) S has the Menger property. (3) For a Lindelöf subspace of a real GO-space (for instance the Sorgenfrey line), total paracompactness, total metacompactness and the Menger property are equivalent.

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Universally meager sets, II

Cited by 14 publications

References 6 publications

Uniform Algebras in the Cantor and Baire Spaces

Uniform Algebras in the Cantor and Baire Spaces

Dirichlet sets, Erdős-Kunen-Mauldin theorem, and analytic subgroups of the reals

Menger subsets of the Sorgenfrey line

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