Properness of foliations. (English) Zbl 1407.57021
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 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.
Reviewer: David B. Gauld (Auckland)
MSC:
| 57R30 | Foliations in differential topology; geometric theory |
| 37B35 | Gradient-like behavior; isolated (locally maximal) invariant sets; attractors, repellers for topological dynamical systems |
| 54D10 | Lower separation axioms (\(T_0\)–\(T_3\), etc.) |
Citations:
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