Journal of Symbolic Logic

$U$-Monad Topologies of 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. The $U$-monad topology of $\mathscr{H}$ is the quotient topology of the $U$-topological space $\mathscr{H}$ modulo $U$. In this paper we answer a question of Keisler and Leth about the $U$-monad topologies by showing that when $\mathscr{H}$ is $\kappa$-saturated and has cardinality $\kappa$, (1) if the coinitiality of $U_1$ is uncountable, then the $U_1$-monad topology and the $U_2$-monad topology are homeomorphic iff both $U_1$ and $U_2$ have the same coinitiality; and (2) $\mathscr{H}$ can produce exactly three different $U$-monad topologies (up to homeomorphism) for those $U$'s with countable coinitiality. As a corollary $\mathscr{H}$ can produce exactly four different $U$-monad topologies if the cardinality of $\mathscr{H}$ is $\omega_1$.

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J. Symbolic Logic, Volume 57, Issue 2 (1992), 534-539.

First available in Project Euclid: 6 July 2007

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Jin, Renling. $U$-Monad Topologies of Hyperfinite Time Lines. J. Symbolic Logic 57 (1992), no. 2, 534--539.

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