  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.
