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Menger’s and Hurewicz’s problems: solutions from “The Book” and refinements. (English) Zbl 1252.03111

Babinkostova, L. (ed.) et al., Set theory and its applications. Annual Boise extravaganza in set theory, Boise, ID, USA, 1995–2010. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-4812-8/pbk). Contemporary Mathematics 533, 211-226 (2011).
Summary: 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 are concrete uncountable sets of reals \(X\) (indeed, \(|X|=\mathfrak b\)), which have the following property:
Given point-cofinite covers \(\mathcal {U}_1\), \(\mathcal{U}_2\), \(\ldots\) of \(X\), there are for each \(n\) sets \(U_n, V_n \in \mathcal{U}_n\), such that each member of \(X\) is contained in all but finitely many of the sets \(U_1 \cup V_1\), \(U_2 \cup V_2\), \(\dots\).
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 \(\mathcal{U}_n\).
For the entire collection see [Zbl 1205.03004].

MSC:

03E17 Cardinal characteristics of the continuum
26A03 Foundations: limits and generalizations, elementary topology of the line
03E75 Applications of set theory
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