Boaz Tsaban, Lyubomyr Zdomskyy
We establish a surprising connection between Menger's classical covering property and Blass's modern combinatorial notion of groupwise density. This connection implies a short proof of the groupwise density bound on the additivity number for Menger's property.
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The main results of this note are: It is consistent that every subparacompact space $X$ of size $ω_1$ is a $D$-space; If there exists a Michael space, then all productively Lindelöf spaces have the Menger property, and, therefore, are $D$-spaces; and Every locally $D$-space which admits a $σ$-locally finite cover by Lindelöf spaces is a $D$-space.
In this note we present a ZFC construction of a non-meager filter which fails to be countable dense homogeneous. This answers a question of Hernández-Gutiérrez and Hrušák. The method of the proof also allows us to obtain a metrizable Baire topological group which is strongly locally homogeneous but not countable dense homogeneous.
Totally paracompact spaces and the Menger covering property
针对正则Lindelöf空间中全仿紧性与Menger覆盖性质是否等价的问题，研究证明二者在实数集上的Lindelöf GO-空间中等价，给出正则Menger空间全仿紧的博弈论证明，还在𝔟=ω₁条件下强化了Michael的相关经典结论。
We prove that in the Miller model the Menger property is preserved by finite products of metrizable spaces. This answers several open questions and gives another instance of the interplay between classical forcing posets with fusion and combinatorial covering properties in topology.
Some properties defined by relative versions of star-covering properties II
本文研究集合星/强星Menger、Hurewicz型相对拓扑性质，证明其与可数紧性、extent的关系，构造反例解答相关公开问题。
We study products of general topological spaces with Menger's covering property, and its refinements based on filters and semifilters. To this end, we extend the projection method from the classic real line topology to the Michael topology. Among other results, we prove that, assuming \CH{}, every productively Lindelöf space is productively Menger, and every productively Menger space is productively Hurewicz. None of these implications is reversible.
A remark on Fremlin-Miller theorem concerning the Menger property and Michael concentrated sets
本文通过 Michael 技巧轻微加强了 [JMSS] 的结果，并在 ZFC 中首次构造出具有 Menger 覆盖性质但不具有 Hurewicz 覆盖性质的实数集，从而解决了 Hurewicz 问题。
We study the preservation of selective covering properties, including classic ones introduced by Menger, Hurewicz, Rothberger, Gerlits and Nagy, and others, under products with some major families of concentrated sets of reals. Our methods include the projection method introduced by the authors in an earlier work, as well as several new methods. Some special consequences of our main results are (definitions provided in the paper): \be \item Every product of a concentrated space with a Hurewicz $\sone(\Ga,\Op)$ space satisfies $\sone(\Ga,\Op)$. On the other hand, assuming \CH{}, for each Sierpi…
Which Isbell--Mrówka spaces ($Ψ$-spaces) satisfy the star version of Menger's and Hurewicz's covering properties? Following Bonanzinga and Matveev, this question is considered here from a combinatorial point of view. An example of a $Ψ$-space that is (strongly) star-Menger but not star-Hurewicz is obtained. The PCF-theory function $κ\mapsto\cof([κ]^\alephes)$ is a key tool. Using the method of forcing, a complete answer to a question of Bonanzinga and Matveev is provided. The results also apply to the mentioned covering properties in the realm of Pixley--Roy spaces, to the extent of spaces wit…
We provide simplified solutions of Menger's and Hurewicz's problems and conjectures, concerning generalizations of sigma-compactness. The reader who is new to this field will find a self-contained treatment in Sections 1, 2, and 5. Sections 3 and 4 contain new results, based on the mentioned simplified solutions. The main new result is that there is a set of reals X of cardinality equal to the unbounding number b, and which has the following property: "Given point-cofinite covers U_1,U_2,... of X, there are for each n sets u_n,v_n in U_n, such that each member of X is contained in all but fini…
Iwasa investigated the preservation of various covering properties of opological spaces under Cohen forcing. By improving the argument in Iwasa's paper, we prove that the Rothberger property, the Menger property and selective screenability are also preserved under Cohen forcing and forcing with the measure algebra.
Combinatorial covering properties in an uncountable setting: canonical examples
本文通过推广经典可数构造，在不可数设定下构造了满足Hurewicz、Menger和γ-性质的空间例子，并揭示了多个新开问题。
On covering properties of end and ray spaces
本文研究了终点空间和射线空间的覆盖性质，证明了σ-紧性和Menger性质等特征的等价性，并为具有可数多个终点的图提供了组合特征。
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 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".
Coideals as remainders of groups distinguishing between combinatorial covering properties
研究通过构造 $2^\omega$ 中具 Menger 余项但缺乏更强组合覆盖性质的子群相容例子，厘清不同覆盖性质，并解答相关开放问题。
The_additivity_number_ of a topological property (relative to a given space) is the minimal number of subspaces with this property whose union does not have the property. The most well-known case is where this number is greater than Aleph_0, i.e. the property is sigma-additive. We give a rather complete survey of the known results about the additivity numbers of a variety of topological covering properties, including those appearing in the Scheepers diagram (which contains, among others, the classical properties of Menger, Hurewicz, Rothberger, and Gerlits-Nagy). Some of the results proved her…
In this paper, we prove the following Theorems 1. An extremally disconnected space $X$ has the semi-Menger property if and only if One does not have a winning strategy in the game $G_{fin}(sO,sO)$. 2. An extremally disconnected space $X$ has the semi-Rothberger property if and only if One does not have a winning strategy in the game $G_1(sO,sO)$.
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。
This article is devoted to the interplay between forcing with fusion and combinatorial covering properties. We illustrate this interplay by proving that in the Laver model for the consistency of the Borel's conjecture, the product of any two metrizable spaces with the Hurewicz property has the Menger property.
Productively Scheepers spaces and their relatives
在假设$\mathfrak{b}=\mathfrak{d}$或$\mathfrak{d}=\aleph_1$的情况下，生成性Scheepers空间在特定条件下等价于生成性Hurewicz空间，并且Scheepers性质与由超滤子参数化的Menger性质等价。
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