Hereditary topological diagonalizations and the Menger-Hurewicz Conjectures

A diagonalization property between Hurewicz and Menger

In classical works, Hurewicz and Menger introduced two diagonalization properties for sequences of open covers. Hurewicz found a combinatorial characterization of these notions in terms of continuous images. Recently, Scheepers has shown that these notions are particular cases in a large family of diagonalization schemas. One of the members of this family is weaker than the Hurewicz property and stronger than the Menger property, and it was left open whether it can be characterized combinatorially in terms of continuous images. We give a positive answer. This paper can serve as an exposition o…

论 $\mathfrak b=\mathfrak d$ 模型中可积 Menger 空间与可积 Hurewicz 空间之间的相互作用

On the interplay between productively Menger and productively Hurewicz spaces in models of $\mathfrak b=\mathfrak d$

本文研究了 Cantor 空间中可积 Menger 与可积 Hurewicz 子空间的关系，证明了在 Laver 模型中两者等价并刻画 Hurewicz 空间，同时在 CH 下两者不等价。

Menger's and Hurewicz's Problems: Solutions from "The Book" and refinements

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…

Scales, fields, and a problem of Hurewicz

Menger's basis property is a generalization of $σ$-compactness and admits an elegant combinatorial interpretation. We introduce a general combinatorial method to construct non $σ$-compact sets of reals with Menger's property. Special instances of these constructions give known counterexamples to conjectures of Menger and Hurewicz. We obtain the first explicit solution to the Hurewicz 1927 problem, that was previously solved by Chaber and Pol on a dichotomic basis. The constructed sets generate nontrivial subfields of the real line with strong combinatorial properties, and most of our results c…

Menger remainders of topological groups

In this paper we discuss what kind of constrains combinatorial covering properties of Menger, Scheepers, and Hurewicz impose on remainders of topological groups. For instance, we show that such a remainder is Hurewicz if and only it is $σ$-compact. Also, the existence of a Scheepers non-$σ$-compact remainder of a topological group follows from CH and yields a $P$-point, and hence is independent of ZFC. We also make an attempt to prove a dichotomy for the Menger property of remainders of topological groups in the style of Arhangel'skii.

Pointless proofs of the Menger and Rothberger games

This work presents proofs for Hurewicz's and Pawlikowski's Theorems about topological games in the context of lattices, in which one cannot argue with points.