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Scales, fields, and a problem of Hurewicz. (English) Zbl 1159.03030

A general combinatorial method for constructing examples of sets of reals with the Menger property, the Hurewicz property and related covering properties defined in terms of selection principles and partition relations is presented. Particular instances of the construction coincide with the classical constructions of non-\(\sigma\)-compact sets of reals with the Menger property or with the Hurewicz property.
Another application gives one of the main results of the paper: a ZFC example of a Menger set that is not Hurewicz (answering a question of Hurewicz from 1927). Although this question had been previously answered by Chaber and Pol, their proof applied a dichotomy: one example was constructed under the assumption \({\mathfrak b}<{\mathfrak d}\) and another from \({\mathfrak b}={\mathfrak d}\). The method is flexible enough to obtain examples with rich algebraic structure. For example, a subfield of \({\mathbb R}\) is constructed which is not Hurewicz, but every finite power is Menger. Many other interesting examples are constructed.

MSC:

03E05 Other combinatorial set theory
03E75 Applications of set theory
54D20 Noncompact covering properties (paracompact, Lindelöf, etc.)
54G20 Counterexamples in general topology
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