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

TL;DR

This paper offers simplified solutions to Menger's and Hurewicz's problems related to sigma-compactness, introduces new results about special sets of reals, and refines understanding of covering properties in topology.

Contribution

It provides simplified solutions to longstanding problems and presents a new set of reals with a stronger covering property than previously known.

Findings

Existence of a set of reals with a specific strong covering property

The new property is strictly stronger than Hurewicz's property

Cannot prove the same result with single set selections from covers

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 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 finitely many of the sets u_1 union v_1,u_2 union v_2,..." 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 U_n.