Journal of Symbolic Logic

$U$-Lusin Sets in Hyperfinite Time Lines

Renling Jin

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In an $\omega_1$-saturated nonstandard universe a cut is an initial segment of the hyperintegers which is closed under addition. Keisler and Leth in [KL] introduced, for each given cut $U$, a corresponding $U$-topology on the hyperintegers by letting $O$ be $U$-open if for any $x \in O$ there is a $y$ greater than all the elements in $U$ such that the interval $\lbrack x - y, x + y\rbrack \subseteq O$. Let $U$ be a cut in a hyperfinite time line $\mathscr{H}$, which is a hyperfinite initial segment of the hyperintegers. A subset $B$ of $\mathscr{H}$ is called a $U$-Lusin set in $\mathscr{H}$ if $B$ is uncountable and for any Loeb-Borel $U$-meager subset $X$ of $\mathscr{H}, B \cap X$ is countable. Here a Loeb-Borel set is an element of the $\sigma$-algebra generated by all internal subsets of $\mathscr{H}$. In this paper we answer some questions of Keisler and Leth about the existence of $U$-Lusin sets by proving the following facts. (1) If $U = x/\mathbb{N} = \{y \in \mathscr{H}: \forall n \in \mathbb{N}(y < x/n)\}$ for some $x \in \mathscr{H}$, then there exists a $U$-Lusin set of power $\kappa$ if and only if there exists a Lusin set of the reals of power $\kappa$. (2) If $U \neq x/\mathbb{N}$ but the coinitiality of $U$ is $\omega$, then there are no $U$-Lusin sets if CH fails. (3) Under ZFC there exists a nonstandard universe in which $U$-Lusin sets exist for every cut $U$ with uncountable cofinality and coinitiality. (4) In any $\omega_2$-saturated nonstandard universe there are no $U$-Lusin sets for all cuts $U$ except $U = x/\mathbb{N}$.

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J. Symbolic Logic, Volume 57, Issue 2 (1992), 528-533.

First available in Project Euclid: 6 July 2007

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Jin, Renling. $U$-Lusin Sets in Hyperfinite Time Lines. J. Symbolic Logic 57 (1992), no. 2, 528--533.

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