Compactness of Loeb Spaces

Abstract

In this paper we show that the compactness of a Loeb space depends on its cardinality, the nonstandard universe it belongs to and the underlying model of set theory we live in. In section 1 we prove that Loeb spaces are compact under various assumptions, and in section 2 we prove that Loeb spaces are not compact under various other assumptions. The results in section 1 and section 2 give a quite complete answer to a question of D. Ross.