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

Abstract

We provide simplified solutions of Menger's and Hurewicz's problems and conjectures, concerning generalizations of σ-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 are concrete uncountable sets of reals X (indeed, |X| = b), which have the following property:Given point-cofinite covers U 1 , U 2 , . . . of X, there are for each n sets Un, Vn ∈ Un, such that each member of X is contained in all but finitely many of the sets. . This property is strictly stronger than Hurewicz's covering property. Miller and the present author showed that one cannot prove the same result if we are only allowed to pick one set from each Un. Dedicated to Professor GideonSchechtman Contents 1. Menger's Conjecture 1 2. Hurewicz's Conjecture 6 3. Strongly Hurewicz sets of reals, in ZFC 9 4. A visit at the border of ZFC 11 5. The Hurewicz Problem 12 Acknowledgments 15 References 15 Appendix A. S f (A , B)