U-lusin sets in hyperfinite time lines

Journal of Symbolic Logic 57 (2):528-533 (1992)
  Copy   BIBTEX


In an ω1-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 ∈ 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 H, which is a hyperfinite initial segment of the hyperintegers. A subset B of H is called a U-Lusin set in H if B is uncountable and for any Loeb-Borel U-meager subset X of H, B ∩ X is countable. Here a Loeb-Borel set is an element of the σ-algebra generated by all internal subsets of 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 ∈ H, then there exists a U-Lusin set of power κ if and only if there exists a Lusin set of the reals of power κ. (2) If U ≠ x/N but the coinitiality of U is ω, 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 ω2-saturated nonstandard universe there are no U-Lusin sets for all cuts U except U = x/N



    Upload a copy of this work     Papers currently archived: 76,479

External links

Setup an account with your affiliations in order to access resources via your University's proxy server

Through your library

Similar books and articles

Meager sets on the hyperfinite time line.H. Jerome Keisler & Steven C. Leth - 1991 - Journal of Symbolic Logic 56 (1):71-102.
U-monad topologies of hyperfinite time lines.Renling Jin - 1992 - Journal of Symbolic Logic 57 (2):534-539.
Existence of some sparse sets of nonstandard natural numbers.Renling Jin - 2001 - Journal of Symbolic Logic 66 (2):959-973.
Set theoretic properties of Loeb measure.Arnold W. Miller - 1990 - Journal of Symbolic Logic 55 (3):1022-1036.
Lusin-sierpinski index for the internal sets.Boško Živaljević - 1992 - Journal of Symbolic Logic 57 (1):172 - 178.
Type two cuts, bad cuts and very bad cuts.Renling Jin - 1997 - Journal of Symbolic Logic 62 (4):1241-1252.
Cuts in hyperfinite time lines.Renling Jin - 1992 - Journal of Symbolic Logic 57 (2):522-527.


Added to PP

28 (#419,425)

6 months
1 (#455,463)

Historical graph of downloads
How can I increase my downloads?

Citations of this work

No citations found.

Add more citations

References found in this work

Model Theory.Michael Makkai, C. C. Chang & H. J. Keisler - 1991 - Journal of Symbolic Logic 56 (3):1096.
Foundations of Infinitesimal Stochastic Analysis.K. D. Stroyan - 1988 - Journal of Symbolic Logic 53 (4):1261-1262.

Add more references