  We prove that if F is a foliation of a compact manifold M with all leaves
compact submanifolds, and the transverse saturated category of F is finite,
then the leaf space M/F is compact Hausdorff. The proof is surprisingly
delicate, and is based on some new observations about the geometry of compact
foliations. Colman proved in her 1998 doctoral thesis that the transverse
saturated category of a compact Hausdorff foliation is always finite, so we
obtain a new characterization of the compact Hausdorff foliations among the
compact foliations as those with finite transverse saturated category.
