Strongly sequentially separable function spaces, via selection principles

Osipov, Alexander V.; Szewczak, Piotr; Tsaban, Boaz

doi:10.1016/j.topol.2019.106942

Cited by 5 publications

(5 citation statements)

References 19 publications

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“…The equivalence was proved by Gerlitz and Nagy [13] for open covers i.e., for S 1 (Ω, Γ), but the proof works also for Borel covers as was noted in Scheepers and Tsaban [35]. In [28], authors was proved Corollary 4.22. For a Tychonoff space X the following statements are equivalent:…”

Section: Continuous Functionsmentioning

confidence: 83%

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Application of selection principles in the study of the properties of function spaces

Osipov

2018

Acta Math. Hungar.

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For a Tychonoff space X, we denote by C p (X) the space of all real-valued continuous functions on X with the topology of pointwise convergence.In this paper we prove that:• if every finite power of X is Lindelëof then C p (X) to be strongly sequentially separable iff X is γ-set.• B α (X) -functions Baire class α (1 < α ≤ ω 1 ) on a Tychonoff space X with the pointwise topology -to be sequentially separable iff there exists a Baire isomorphism class α from a space X onto a σ-set.• B α (X) is strongly sequentially separable iff iw(X) = ℵ 0 and X is a Z α -cover γ-set for 0 < α ≤ ω 1 .• there is a consistent example of a set of reals X, such that C p (X) is strongly sequentially separable, but B 1 (X) is not strongly sequentially separable.• B(X) is sequentially separable, but is not strongly sequentially separable for a b-Sierpiński set X.

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Section: Continuous Functionsmentioning

confidence: 83%

“…In [28], authors was proved Theorem 3.2. For a Tychonoff space X the following statements are equivalent:…”

Section: Continuous Functionsmentioning

confidence: 99%

Application of selection principles in the study of the properties of function spaces

Osipov

2018

Acta Math. Hungar.

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“…We also weaken a set-theoretic assumption in the result of Bartoszyński and Recław. In both the cases we use combinatorial methods of construction of subsets of with the property invented by Tsaban ([7, Theorem 3.6], [8, Theorem 6]) and developed by Włudecka and the first named author [13]. We also use a combinatorial covering characterization of null-additive subsets of given by Zindulka [17].…”

Section: Introductionmentioning

confidence: 99%

Null Sets and Combinatorial Covering Properties

Szewczak¹,

Weiss²

2021

J. symb. log.

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A subset of the Cantor cube is null-additive if its algebraic sum with any null set is null. We construct a set of cardinality continuum such that: all continuous images of the set into the Cantor cube are null-additive, it contains a homeomorphic copy of a set that is not null-additive, and it has the property γ, a strong combinatorial covering property. We also construct a nontrivial subset of the Cantor cube with the property γ that is not null additive. Set-theoretic assumptions used in our constructions are far milder than used earlier by Galvin-Miller and Bartoszy ński-Recław, to obtain sets with analogous properties. We also consider products of Sierpi ński sets in the context of combinatorial covering properties. §1. Introduction. Let N be the set of natural numbers and P(N) be the power set of N. We identify each set in P(N) with its characteristic function, an element of the Cantor cube {0, 1} N ; in that way we introduce topology in P(N). The Cantor space P(N) with the symmetric difference operation ⊕ is a topological group; this operation coincides with the addition modulo 2 in {0, 1}In an analogous way, define null-additive subsets of the real line with the addition + as a group operation. As we see in the forthcoming Theorem 2.2, it is relatively consistent with ZFC that null-additive subsets of P(N) are not preserved by homeomorphisms into P(N). Subsets of the real line whose all continuous images into the real line are null-additive were considered by Galvin and Miller [3]; to this end they used combinatorial covering properties.By space we mean a Tychonoff topological space. A cover of a space is a family of proper subsets of the space whose union is the entire space. An open cover of a space is a cover whose members are open subsets of the space. A cover of a space is an -cover if each finite subset of the space is contained in a set from the cover and it is a γ-cover if it is infinite and each point of the space belongs to all but finitely many sets from the cover. A space has the property γ if every open -cover of the space contains a γ-cover. This property was introduced by Gerlits and Nagy in the context of local properties of functions spaces [4]. They proved that a space X has the property γ if and only if the space C p (X ) of all continuous real-valued functions defined on X with the pointwise convergence topology is Fréchet-Urysohn, i.e., each point in the closure of a subset of C p (X ) is a limit of a sequence from the set [4, Theorem 2]. Galvin and Miller observed that for a subset of the real line X with the property γ and a meager subset of the real line Y, the set

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“…We also weaken a set-theoretic assumption in the result of Bartoszyński and Rec law. In the both cases we use combinatorial methods of construction of subsets of P(N) with the property γ invented by Tsaban ([10, Theorem 3.6], [11,Theorem 6]) and developed by W ludecka and the first named author [16]. We also use a combinatorial covering characterization of null-additive subsets of P(N) given by Zindulka [20].…”

Section: Introductionmentioning

confidence: 99%

Null sets and combinatorial covering properties

Szewczak,

Weiss

2020

Preprint

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Strongly sequentially separable function spaces, via selection principles

Cited by 5 publications

References 19 publications

Application of selection principles in the study of the properties of function spaces

Application of selection principles in the study of the properties of function spaces

Null Sets and Combinatorial Covering Properties

Null sets and combinatorial covering properties

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