Properness of foliations (Q1710647)

It is shown that a continuous foliation \(\mathcal F\) on a metrisable manifold \(M\) is proper if and only if the leaf space \(M/\mathcal F\) is \(T_0\). An example of \textit{M. Kneser} [Arch. Math. 11, 280--281 (1960; Zbl 0102.38601)], is recalled of a single leaf codimension one foliation on a non-metrisable 3-manifold that shows that the hypothesis of metrisability is required. A related result for orbits of a continuous flow on a metrisable manifold is presented.