Products of general Menger spaces

Szewczak, Piotr; Tsaban, Boaz

doi:10.1016/j.topol.2019.01.005

“…Assuming , in the class of sets of reals, no Luzin set is productively [11, Corollary 2.11]. There are open problems ([11, Problem 7.5], [12, Problem 5.5]), whether in the class of sets of reals (or in the class of general topological spaces) for any Sierpiński set X , there is a space Y satisfying such that the product space does not satisfy , and what if we assume the Continuum Hypothesis? We consider an analogous problem with respect to combinatorial covering properties, stronger than .…”

Section: Nonproductivity Of Sierpiński-type Setsmentioning

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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“…A space is productively P if its product space with any space satisfying P, satisfies P. In the class of sets of reals, no Luzin set is productively S fin (O, O). There are open problems ([14, Problem 7.5], [15,Problem 5.5]), whether in the class of sets of reals (or in the class of general topological spaces) for any Sierpiński set X, there is a space Y satisfying U fin (O, Γ) such that the product space X × Y does not satisfy U fin (O, Γ), and what if we assume the Continuum Hypothesis? We consider an analogous problem with respect to combinatorial covering properties, stronger than U fin (O, Γ).…”

Section: Nonproductivity Of Sierpiński-type Setsmentioning

confidence: 99%

Null sets and combinatorial covering properties

Szewczak,

Weiss

2020

Preprint

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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 Bartoszyński-Rec law, to obtain sets with analogous properties. We also consider products of Sierpiński sets in the context of combinatorial covering properties.

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“…In general topology, several topological properties are not finitely productive, such as paracompactness, strong paracompactness, Lindelöfness, and metacompactness. e area of research regarding the problem "What conditions on (Y, σ) and (Z, δ) to insure that their product has property P"is still hot [38][39][40][41][42][43][44][45]. e second goal of this paper is to introduce several product theorems concerning metacompactness.…”

Section: Introductionmentioning

confidence: 99%

Theta Omega Topological Operators and Some Product Theorems

Ghour

¹

,

El-Issa

²

2021

Mathematical Problems in Engineering

0

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We introduce and investigate the concepts of θ ω -limit points and θ ω -interior points, and we use them to introduce two new topological operators. For a subset B of a topological space Y , σ , denote the set of all limit points of B (resp. θ -limit points of B , θ ω -limit points of B , interior points of B , θ -interior points of B , and θ ω -interior points of B ) by D B (resp. D θ B , D θ ω B , Int B , Int θ B , and Int θ ω B ). Several results regarding the two new topological operators are given. In particular, we show that D θ ω B lies strictly between D B and D θ B and Int θ ω B lies strictly between Int θ B and Int B . We show that D B = D θ ω B (resp. Cl θ B = Cl θ ω B and D B = D θ ω B = D θ B ) for locally countable topological spaces (resp. antilocally countable topological spaces and regular topological spaces). In addition to these, we introduce several product theorems concerning metacompactness.

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Products of general Menger spaces

Cited by 8 publications

References 19 publications

Null Sets and Combinatorial Covering Properties

Null Sets and Combinatorial Covering Properties

Null sets and combinatorial covering properties

Theta Omega Topological Operators and Some Product Theorems

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