Properness of foliations

Abstract

In this paper, we consider the question whether the leaf space of a foliation on a manifold is T0 if and only if each leaf is proper. In general, the answer is negative because there is a codimension one foliation of a non-paracompact 3-dimensional manifold constructed by Milnor which consists of only one leaf, which is non-proper. However, using the notions of dynamical systems, we show that the answer for paracompact manifolds is affirmative. In other words, a foliation F on a paracompact manifold M is proper if and only if the leaf space M/F is T0. In fact, a leaf L of a foliation on a paracompact manifold is proper if and only if its leaf class (i.e. the subset of points whose leaf closures coincide with the closure of L) consists of only L itself. To show this, we show that the subset of leaves contained in the leaf class of a non-proper leaf L∈F of a foliation F on a paracompact manifold for has the cardinality of the continuum. In addition, similar results hold for flows and homeomorphisms.