U-Monad Topologies of 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 hyper nite time lineH, which is a hyper nite initial segment of the hyperintegers. The U monad topology of H is the quotient topology of the U topological space H modulo U . In this paper we answer a question of Keisler and Leth about the U monad topologies by showing that when H is saturated and has cardinality , (1) if the coinitiality of U1 is uncountable, then the U1 monad topology and the U2 monad topology are homeomorphic i both U1 and U2 have the same coinitiality; (2) H can produce exactly three di erent U monad topologies (up to homeomorphism) for those U 's with countable coinitiality. As a corollary H can produce exactly four di erent U monad topologies if the cardinality of H is !1 . 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. Let us recall that a cut in H is an initial segment of H which is closed under addition. A cut must be external. Let U be a cut in H. A subset O of H is called U open if for any x 2 O there is a y 2 H U such that [x y; x + y] O. All U open sets form a U topology on H. Let U be a cut in H and x 2 H. x=U = fy 2 H : 8z 2 U (y x=z)g, which is also a cut. cf(U), the co nality of U , = minfcard(F ) : F U and F is co nal in Ug. ci(U), the coinitiality of U , = minfcard(F ) : F H U and F is coinitial in H Ug. Let U be a cut in H. For each x 2 H we let U monad(x), the U monad of x, = fy 2 H : jy xj 2 Ug. For a subset B of H U monad(B)= fU monad(x) : x 2 Bg. By a U monad we mean a U monad(x) for some x 2 H.