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- W3148568330 abstract "Abtract We prove that every countably determined set C is U -meager if and only if every internal subset A of C is U -meager, provided that the cofinality and coinitiality of the cut U are both uncountable. As a consequence we prove that for such cuts a countably determined set C which intersects every U -monad in at most countably many points is U -meager. That complements a similar result in [KL]. We also give some partial solutions to some open problems from [KL]. We prove that the set , where H is an infinite integer, cannot be expressed as a countable union of countably determined sets each of which is U -meager for some cut U with min{cf( U ), ci( U )} ≥ ω 1 . Also, every Borel, or countably determined set C which is U -meager for every cut U is a countable union of Borel, or countably determined sets respectively, which are U -nowhere dense for every cut U . Further, the class of Borel U -meager sets for min{cf( U ),ci( U )} ≥ ω 1 coincides with the least family of sets containing internal U -meager sets and closed with respect to the operation of countable union and intersection. The same is true if the phrase “ U -meager sets” is replaced by “ U -meager for every cut U .”" @default.
- W3148568330 created "2021-04-13" @default.
- W3148568330 creator A5088197740 @default.
- W3148568330 date "1991-09-01" @default.
- W3148568330 modified "2023-10-14" @default.
- W3148568330 title "<i>U</i>-meager sets when the cofinality and the coinitiality of <i>U</i> are uncountable" @default.
- W3148568330 cites W2315435043 @default.
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- W3148568330 doi "https://doi.org/10.2307/2275060" @default.
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