U-Lusin Sets in Hyperfinite Time Lines

In an !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 2 O there is a y greater than all the elements in U such that the interval [x y; x+y] O. Let U be a cut in a hypernite time line H, which is a hyper nite initial segment of the hyperintegers. A subsetB ofH is called a U Lusin set inH ifB is uncountable and for any Loeb{ Borel U meager subset X of H, B T X is countable. Here a Loeb{Borel set is an element of the {algebra generated by all internal subsets ofH. In this paper we answer some questions of Keisler and Leth about the existence of U Lusin sets by proving that: (1) If U = x=N = fy 2 H : 8n 2 N (y <x=n)g for some x 2 H, then there exists a U Lusin set of power if and only if there exists a Lusin set of the reals of power ; (2) If U 6= x=N but the coinitiality of U is !, 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 co nality and coinitiality; (4) In any !2 saturated nonstandard universe there are no U Lusin sets for all cuts U except U = x=N . Throughout this paper we work within !1 saturated nonstandard universes. We let M be a nonstandard universe and N be the set of all hyperintegers in M which contains N , the set of all standard positive integers. Let H 2 N N ; we call H = fn 2 N : n H g a hyper nite time line or a hyperline for short. We always let H be the largest element of H. Let [a; b] = fx 2 H : a x bg be an interval in H and [r] = maxfn 2 N : n rg for any hyperreal r. A notion of Loeb measure for H, which is the standard part of the countably additive extension of the counting measure on H, was introduced by P. Loeb (cf.[Lo]) as a counterpart of Lebesgue measure for the reals. Recently H. J. Keisler and S. Leth (cf.[KL]) introduced U topologies on H for any cuts U as an analogue of the order topology on the reals. They discussed the relationship between U meager sets and Loeb measure zero sets and the existence of Loeb{Sierpi nski sets and U Lusin sets. They listed many questions at the end of the paper. In this paper we discuss some of those questions about the existence of U Lusin sets. Most of the questions discussed here were motivated by the results of [KL], [M1] and [M2]. For background in model