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Additivity properties of topological diagonalizations. (English) Zbl 1071.03031

In 1996, W. Just, A. W. Miller, M. Scheepers and P. J. Szeptycki [Topology Appl. 73, 241–266 (1996; Zbl 0870.03021)] introduced a general framework for so-called “selection principles”. This framework unified a number of properties of sets of reals (Hurewicz sets, Menger sets, \(\gamma\)-sets of Gerlits and Nagy etc.) and raised the question, which of the properties are closed under finite or countable unions. In the paper under review the authors study this question. The paper begins with a short survey indicating for which classes of spaces the question has been settled (it was known that CH implies \(S_1(\Omega,\Gamma)\) is not finitely additive and that \(S_1(\Gamma,\Gamma),\;S_1({\mathcal O},{\mathcal O}),\;S_1(\Gamma,{\mathcal O})\), \(\bigcup_{\text{fin}}(\Gamma,\Gamma)\) and \(\bigcup_{\text{fin}}(\Gamma,{\mathcal O})\) are provably countably additive). The authors’ main results include:
1. If \(2^\omega\) is not the union of \(<2^{\aleph_0}\) meager sets, then there are sets of reals \(X_1, X_2\in S_1(\Omega,\Omega)\) such that \(X_1\cup X_2\not\in \bigcup_{\text{fin}}(\Gamma,\Omega)\). (Hence the classes \(S_1(\Omega,\Omega)\), \(S_1(\Gamma,\Omega)\), \(S_{\text{fin}}(\Omega,\Omega)\), \(S_{\text{fin}}(\Gamma,\Omega)\), \(\bigcup_{\text{fin}}(\Gamma,\Omega)\) are consistently not closed under finite unions).
2. Assuming the near coherence of filters (NCF), \(\bigcup_{\text{fin}}(\Gamma,\Omega)\) is countably additive. In addition, the authors consider the additivity of the Borel versions of some selection principles, proving the analogue to item 1 above for Borel covers. An extended version of the paper with further results and more detailed proofs is available online at //arxiv.org/abs/math.LO/0112262.

MSC:

03E05 Other combinatorial set theory
03E04 Ordered sets and their cofinalities; pcf theory
03E35 Consistency and independence results
03E75 Applications of set theory
54D20 Noncompact covering properties (paracompact, Lindelöf, etc.)
54H05 Descriptive set theory (topological aspects of Borel, analytic, projective, etc. sets)

Citations:

Zbl 0870.03021
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References:

[1] T. Bartoszynski, S. Shelah, and B. Tsaban Additivity properties of topological diagonalizations , (full version). Available at http://arxiv.org/abs/math.LO/0112262. · Zbl 1071.03031 · doi:10.2178/jsl/1067620185
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[5] W. Just, A. W. Miller, M. Scheepers, and P. J. Szeptycki The combinatorics of open covers II , Topology and its Applications , vol. 73 (1996), pp. 241–266. · Zbl 0870.03021 · doi:10.1016/S0166-8641(96)00075-2
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[8] M. Scheepers and B. Tsaban The combinatorics of Borel covers , Topology and its Applications , vol. 121 (2002), pp. 357–382. · Zbl 1025.03042 · doi:10.1016/S0166-8641(01)00078-5
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