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 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 U1,U2,... of X, there are for each n sets un,vn in Un, such that each member of X is contained in all but finitely many of the sets u1 union v1,u2 union v2,..." This property is strictly stronger than Hurewicz's covering property, and by a result of Miller and the present author, one cannot prove the same result if we are only allowed to pick one set from each Un.