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

Authors
  • Tsaban, Boaz
Type
Preprint
Publication Date
Mar 09, 2017
Submission Date
Sep 30, 2009
Source
arXiv
License
Yellow
External links

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.

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