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Combinatorial images of sets of reals and semifilter trichotomy. (English) Zbl 1158.03033

A semifilter is a nonempty set \(S\subseteq[\mathbb N]^{\aleph_0}\) such that for all \(a,b\in[\mathbb N]^{\aleph_0}\), if \(a\subseteq^*b\) and \(a\in S\) then also \(b\in S\). The semifilter trichotomy is the statement (consistent with ZFC) that for each semifilter \(S\) there is an increasing function \(h\in[\mathbb N]^{\aleph_0}\) such that \(S/h\) is either the Fréchet filter, or an ultrafilter, or \([\mathbb N]^{\aleph_0}\), where \(S/h=\{a/h:a\in S\}\) and \(a/h=\{n:a\cap[h(n),h(n+1)]\neq\emptyset\}\). The authors prove that the semifilter trichotomy implies a negative answer to a slightly stronger form of the Hurewicz problem.

MSC:

03E35 Consistency and independence results
03E65 Other set-theoretic hypotheses and axioms
37F20 Combinatorics and topology in relation with holomorphic dynamical systems
54D20 Noncompact covering properties (paracompact, Lindelöf, etc.)
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