Boaz Tsaban
Motivated by a recent result of Sakai, we define a new selection operator for covers of topological spaces, inducing new selection hypotheses. We initiate a systematic study of the new hypotheses. Some intriguing problems remain open.
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We survey some of the major open problems involving selection principles, diagonalizations, and covering properties in topology and infinite combinatorics. Background details, definitions and motivations are also provided.
In this paper we settle all questions whether (it is consistent that) the properties P and Q [do not] coincide, where P and Q run over selection principles of the type U_fin(O,A).
We give a selection of major open problems involving selective properties, diagonalizations, and covering properties for sets of real numbers. This is a revision of the version published as a chapter in the book \textbf{Open Problems in Topology II} (E. Pearl, ed.), Elsevier B.V., 2007, 91--108. The present version reports solutions of some problems, uses up-to-date notation, and update bibliography. Comments and further updates would be appreciated.
Motivated by the definition of classical star selection principles, Cruz-Castillo, Ramírez-Páramo and Tenorio defined some selection principles and posed several questions about relationships between these notions and some of the classical star selection principles. The main goal of this paper is to answer the questions posed by these authors. In addition, we show that some of these defined selection principles are equivalent by taking collections of complements; also, some other results are provided involving collections of refinements.
We study diagonalizations of covers using various selection principles, where the covers are related to linear quasiorderings (tau-covers). This includes: equivalences and nonequivalences, combinatorial characterizations, critical cardinalities and constructions of special sets of reals. This study leads to a solution of a topological problem which was suggested to the author by Scheepers (and stated in an earlier work) and is related to the Minimal Tower problem. We also introduce a variant of the notion of tau-cover, called tau^*-cover, and settle some problems for this variant which are sti…
We investigate selection principles which are motivated by Arhangel'skiǐ's $α_i$-properties, $i=1,2,3,4$, and their relations with classical selection principles. It will be shown that they are closely related to the selection principle $\sone$ and often are equivalent to it.
In this paper we connect selection principles on a topological space to corresponding selection principles on one of its hyperspaces. We unify techniques and generalize theorems from the known results about selection principles for common hyperspace constructions. This includes results of Lj.D.R. Kočinac, Z. Li, and others. We use selection games to generalize selection principles and we work with strategies of various strengths for these games. The selection games we work with are primarily abstract versions of the selection principles of Rothberger, Menger, and Hurewicz type, as well as game…
Arhangelskii's properties $α_2$ and $α_4$ defined for convergent sequences may be characterized in terms of Scheeper's selection principles. We generalize these results to hold for more general collections and consider these results in terms of selection games.
In this paper we investigate the properties of function spaces using the selection principles.
We develop the semifilter approach to the classical Menger and Hurewicz covering properties and show that the small cardinal g is a lower bound of the additivity number of the family of Menger subspaces of the Baire space, and under u< g every subset X of the real line with the property Split(Lambda,Lambda) is Hurewicz.
We review some selected recent results concerning selection principles in topology and their relations with several topological constructions.
We characterize countable dimensionality and strong countable dimensionality by means of an infinite game.
In this paper we define some combinatorial principles to characterize spaces $X$ whose hyperspace satisfies some variation of some classical star selection principle. Specifically, the variations characterized are the selective and absolute versions of the star selection principles for the Menger and Rothberger cases; also, the hyperspaces considered in these characterizations are $CL(X)$, $\mathbb{K}(X)$, $\mathbb{F}(X)$ and $\mathbb{CS}(X)$ in both cases, endowed with either the Fell topology or the Vietoris topology.
Arhangel'skii proved that if a first countable Hausdorff space is Lindelöf, then its cardinality is at most $2^{\aleph_0}$. Such a clean upper bound for Lindelöf spaces in the larger class of spaces whose points are ${\sf G}_δ$ has been more elusive. In this paper we continue the agenda started in F.D. Tall, On the cardinality of Lindelöf spaces with points $G_δ$, Topology and its Applications 63 (1995), 21 - 38, of considering the cardinality problem for spaces satisfying stronger versions of the Lindelöf property. Infinite games and selection principles, especially the Rothberger property, a…
We continue to investigate applications of $k$-covers in function spaces with the compact-open topology.
This article is devoted to the interplay between forcing with fusion and combinatorial covering properties. We discuss known instances of this interplay as well as present a new one, namely that in the Laver model for the consistency of the Borel's conjecture, the Hurewicz property is preserved by finite products of metrizable spaces.
The product of a Sierpinski set and a Lusin set has Menger's property. The product of a gamma set and a Lusin set has Rothberger's property.
We study some closure-type properties of function spaces endowed with the new topology of strong uniform convergence on a bornology introduced by Beer and Levy in 2009. The study of these function spaces was initiated in [2] and [3]. The properties we study are related to selection principles.
We extend known results of selection principles in $C_p$-theory to the context of spaces of the form $C_{\mathcal{B}}(X)$, where $\mathcal{B}$ is a bornology on $X$. Particularly, by using the filter approach of Jordan to $C_p$-theory, we show that $γ$-productive spaces are productive with a larger class of $γ$-like spaces.
We prove that if X is a separable metric space with the Hurewicz covering property, then the Banach-Mazur game played on X is determined. The implication is not true when "Hurewicz covering property" is replaced with "Menger covering property".
Certain results on selection principles associated with bornological structure in topological spaces
本文研究拓扑空间中与有界覆盖相关的选择原理，证明若干选择原理的等价性，利用基数不变量建立有界结构基的基数与选择原理的联系，并刻画函数空间的紧致性变体。
Homotopy deals with the intuitive idea of continuous deformation of a continuous map between two topological spaces. In this paper, we introduce homotopical selection principles in bitopological dynamical systems. Here, we define $BTDS-$homotopy, $BTDS-$iteration homotopy and $BTDS-$path homotopy, and then using these notions we define various selection properties viz. H-Rothberger property, H-Menger property, PH-Rothberger property, PH-Menger property and their weaker versions. We also discuss several results connecting these concepts in bitopological dynamical systems.
In this paper, we introduce the notions of Star-$σ\mathcal{K}$ and absolutely Star-$σ\mathcal{K}$ spaces which allow us to unify results among several properties in the theory of star selection principles on small spaces. In particular, results on star selective versions of the Menger, Hurewicz and Rothberger properties and selective versions of property $(a)$ regarding the size of the space. Connections to other well-known star properties are mentioned. Furthermore, the absolute and selective version of the neighbourhood star selection principle are introduced. As an application, it is obtain…
On certain localized version of uniform selection principles
本文在一致空间中引入 Menger、Hurewicz 和 Rothberger 选择原理的局部化版本，研究其临界基数、保持性质、蕴含关系及相互作用，并通过例子展示新概念的可区分行为。
Certain observations on selection principles related to bornological covers using ideals
该研究利用理想概念探究有界覆盖相关选择原理，通过基数不变量建立有界基基数与选择原理的关系，推导理想序下选择原理的蕴含关系，证明伪并理想下的选择原理等价性及对应Hurewicz性质的一致性，并用图表呈现相关关系。
This article is a continuation of the study of bornological open covers and related selection principles in metric spaces done in (Chandra et al. 2020) using the idea of strong uniform convergence (Beer and Levi, 2009) on bornology. Here we explore further ramifications, presenting characterizations of various selection principles related to certain classes of bornological covers using the Ramseyan partition relations, interactive results between the cardinalities of bornological bases and certain selection principles involving bornological covers, producing new observations on the $\mathfrak{…
No convenient internal characterization of spaces that are productively Lindelof is known. Perhaps the best general result known is Alster's internal characterization, under the Continuum Hypothesis, of productively Lindelof spaces which have a basis of cardinality at most $\aleph_1$. It turns out that topological spaces having Alster's property are also productively weakly Lindelof. The weakly Lindelof spaces form a much larger class of spaces than the Lindelof spaces. In many instances spaces having Alster's property satisfy a seemingly stronger version of Alster's property and consequently …
Selection principles and proofs from the Book
本文为拟正规收敛定理、Scheeppers 图最后定理、Menger 博弈定理及 Rothberger 覆盖性质可加性下界四个选择原理基本定理提供简化证明，并由此导出若干新结果。
A separable space is strongly sequentially separable if, for each countable dense set, every point in the space is a limit of a sequence from the dense set. We consider this and related properties, for the spaces of continous and Borel real-valued functions on Tychonoff spaces, with the topology of pointwise convergence. Our results solve a problem stated by Gartside, Lo, and Marsh.
We consider the preservation under products, finite powers, and forcing, of a selection principle based covering property of $T_0$ topological groups. Though the paper is in part a survey, it contributes some new information, including: 1. The product of a strictly o-bounded group with an o-bounded group is an o-bounded group - Corollary 18 2. In the generic extension by a finite support iteration of $\aleph_1$ Hechler reals the product of any o-bounded group with a ground model $\aleph_0$ bounded group is an o-bounded group - Theorem 19 3. In the generic extension by a countable support itera…
For a Tychonoff space $X$, we denote by $(C(X), τ_k, τ_p)$ the bitopological space of all real-valued continuous functions on $X$ where $τ_k$ is the compact-open topology and $τ_p$ is the topology of pointwise convergence. In papers [5, 6, 13] variations of selective separability and tightness in $(C(X), τ_k, τ_p)$ were investigated. In this paper we continued to study the selective properties and the corresponding topological games in the space $(C(X), τ_k, τ_p)$.
Further investigations on certain star selection principles
本文进一步研究 Menger、Hurewicz 和 Rothberger 性质的星型版本，利用 Alster 覆盖与临界基数揭示新观察，并借助表格和蕴涵图阐明这些星型选择原理的性质及相互关系。
Let ${\mathcal A}\subset {\mathcal P}(X)$, $\emptyset, X\in {\mathcal A}$, ${\mathcal A}$ being closed under finite intersections. If $ψ={o},ω,γ$, then $Ψ({\mathcal A})$ is the family of those $ψ$-covers ${\mathcal U}$ for which ${\mathcal U}\subseteq {\mathcal A}$. In~\cite{BL2} I have introduced properties $(Ψ_0$ of a~family $F\subseteq {}^XR$ of real functions. The main result of the paper Theorem reads as follows: if~$Φ=Ω,Γ$, then for any couple $\langle Φ,Ψ\rangle$ different from $\langle Ω,{\mathcal O}\rangle$, $X$ has the covering property~{\rm S}${}_1(Φ({\mathcal A}),Ψ({\mathcal A}))$ …
We present a new variation of the classical selection principles $\mathsf{S}_\mathrm{k}(\mathcal A, \mathcal B)$ ($k\in\mathbb N$) and $\mathsf{S}_\mathrm{fin}(\mathcal A, \mathcal B)$ that formally lies between these two properties. As in the case of the classical selection principles, we also obtain a new variation of topological games and discuss how new topological properties may emerge in the specific cases of covering and tightness.
This is the 34th issue of this bulletin, dedicated to selection principles in mathematics. Announcements include, among other things, a call for papers for the Topology and its Applications special issue on selection principles, and the solution of the minimal tower problem in a surprising way.
In this paper, we investigate what selection principles properties are possessed by small (with respect to the bounding and dominating numbers) unions of spaces with certain (star) selection principles.. Furthermore, we give several results about iterations of these properties and weaker properties than paracompactness. In addition, we study the behaviour of these iterated properties on $Ψ$-spaces. Finally, we show that, consistently, there is a normal star-Menger space that is not strongly star-Menger.
Selection principles and the Scheepers diagram in an uncountable setting
本文研究不可数情形下的选择原理，确定了Scheepers图（直至三个未解决蕴含），证明弱紧致κ下κ-Hurewicz猜想在GCH下不成立，并给出相关尺度集与K_κ-空间的新例子。
On certain star-$K$ variant of a selection principle
本文引入并研究星-$K$-Scheepers 性质及其博弈，探讨其与其他 Scheepers 博弈变体的关系，并以蕴含图展示策略关联，同时基于 Alster 覆盖和 Alexandroff 重复提出观察，考察保持性质并列出开放问题。
A space is called linearly H-closed iff any chain cover possesses a dense member. This property lies strictly between feeble compactness and H-closedness. While regular H-closed spaces are compact, there are linearly H-closed spaces which are even collectionwise normal and Fréchet-Urysohn. We give examples in other classes, and ask whether there is a first countable normal linearly H-closed non-compact space in ZFC. We show that PFA implies a negative answer if the space is moreover either locally separable or locally compact and locally ccc. Ostaszewski space (built with $\diamondsuit$) is an…
Let $T\subset\mathbb{R}$ and $(X,\mathcal{U})$ be a uniform space with an at most countable gage of pseudometrics $\{d_p:p\in\mathcal{P}\}$ of the uniformity $\mathcal{U}$. Given $f\in X^T$ (=the family of all functions from $T$ into $X$), the approximate variation of $f$ is the two-parameter family $\{V_{\varepsilon,p}(f):\varepsilon>0,p\in\mathcal{P}\}$, where $V_{\varepsilon,p}(f)$ is the greatest lower bound of Jordan's variations $V_p(g)$ on $T$ with respect to $d_p$ of all functions $g\in X^T$ such that $d_p(f(t),g(t))\le\varepsilon$ for all $t\in T$. We establish the following pointwise…
We present a selection principle $S_1(\mathcal{O},\mathcal{H})$ that characterizes the $G_δ$-diagonal property. We also present a topological game induced by this selection principle and we study the relations between this game and the $G_δ$-property. Finally, we give some applications and examples.
Using the idea of strong uniform convergence on bornology, Caserta, Di Maio and Kočinac studied open covers and selection principles in the realm of metric spaces (associated with a bornology) and function spaces (w.r.t. the topology of strong uniform convergence). We primarily continue in the line initiated before and investigate the behaviour of various selection principles related to these classes of bornological covers. In the process we obtain implications among these selection principles resulting in Scheepers' like diagrams. We also introduce the notion of strong-$\mathfrac{B}$-Hurewicz…
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