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Hereditary topological diagonalizations and the Menger-Hurewicz Conjectures. (English) Zbl 1137.54018

This paper deals with selection/diagonalization properties stating that from the terms of a sequence of covers of type \(\mathcal{A}\) one may select small (one-element/finite) subfamilies to form a cover of type \(\mathcal{B}\). The authors deal with four types of (countable) covers: covers, \(\omega\)-covers (each finite subset is in some element of the cover), \(\tau\)-covers (each element is in infinitely many elements of the cover and if \(x\neq y\) then one of \(x\in U\not\ni y\) or \(x\notin U\ni y\) holds for only finitely many \(U\)) and \(\gamma\)-covers (each point is in all but finitely many members of the cover).
In this paper the authors investigate whether for open and Borel covers these properties are hereditary. A fair number of Borel properties are hereditary; this is because a countable cover by Borel sets of a subspace is easily augmented (using the complement of its union) to form a countable Borel cover of the ambient space.
From \(\mathfrak{p}=\mathfrak{c}\) the authors construct a set of reals with the property that for every sequence of \(\omega\)-covers there is a choice function that forms a \(\gamma\)-cover, yet when the rationals are removed one obtains a space with a sequence \(\langle \mathcal{U}_n\rangle_n\) of \(\gamma\)-covers such that for no finite choices \(\mathcal{F}_n \subseteq \mathcal{U}_n\) the family \(\{\bigcup\mathcal{F}_n\}_n\) is a cover. This shows that none of the properties studied in the paper is provably hereditary.
Reviewer: K. P. Hart (Delft)

MSC:

54D20 Noncompact covering properties (paracompact, Lindelöf, etc.)
54G20 Counterexamples in general topology
54A35 Consistency and independence results in general topology
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