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Strong meager properties for filters. (English) Zbl 0839.03033

The author studies filters \({\mathcal F}\) on the set of natural numbers viewed as subsets of the power set \({\mathcal F}\subseteq {\mathcal P} (\omega)\). This power set is identified with \({}^\omega 2\) (the set of 0-1 sequences) equipped with product topology. Saying that \({\mathcal F}\) has a topological property, we mean that \({\mathcal F}\subseteq {\mathcal P} (\omega) \approx {}^\omega 2\) has the property as a subset of the space \({}^\omega 2\). The author studies various strengthenings of \({\mathcal F}\) being meager, some of them classical, introduced by M. Talagrand [Stud. Math. 74, 283-291 (1982; Zbl 0503.04003)], and some new, introduced in this paper. These results shed more light to our knowledge of filters defined by a MAD-family although the main problem – how to force a new subset below such a filter without adding an unbounded real – is still open.

MSC:

03E05 Other combinatorial set theory
03E35 Consistency and independence results
03E15 Descriptive set theory

Citations:

Zbl 0503.04003
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