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. 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.
"Compact foliations with finite transverse LS category." J. Math. Soc. Japan 70 (3) 1015 - 1046, July, 2018. https://doi.org/10.2969/jmsj/76837683