Abstract

Type two cuts, bad cuts and very bad cuts are introduced in [10] for studying the relationship between Loeb measure and U-topology of a hyperfinite time line in an $\omega_1$-saturated nonstandard universe. The questions concerning the existence of those cuts are asked there. In this paper we answer, fully or partially, some of those questions by showing that: type two cuts exist, the $\aleph_1$-isomorphism property implies that bad cuts exist, but no bad cuts are very bad.