Journal of Symbolic Logic

$U$-Meager Sets when the Cofinality and the Coinitiality of $U$ are Uncountable

Bosko Zivaljevic

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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 $\mathscr{K} = \{1,\ldots,H\}$, 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 $\min\{\mathrm{cf} (U), \mathrm{ci} (U)\} \geq \omega_1$. Also, every Borel, $\Sigma^1_m$ or countably determined set $C$ which is $U$-meager for every cut $U$ is a countable union of Borel, $\Sigma^1_m$ or countably determined sets respectively, which are $U$-nowhere dense for every cut $U$. Further, the class of Borel $U$-meager sets for $\min\{\mathrm{cf}(U), \mathrm{ci}(U)\} \geq \omega_1$ 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 $U$."

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J. Symbolic Logic, Volume 56, Issue 3 (1991), 906-914.

First available in Project Euclid: 6 July 2007

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Zivaljevic, Bosko. $U$-Meager Sets when the Cofinality and the Coinitiality of $U$ are Uncountable. J. Symbolic Logic 56 (1991), no. 3, 906--914. doi:10.2178/jsl/1183743739.

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