Non-Hausdorff Manifolds and Foliations

摘要: Relaxing the Hausdorff condition for a manifold opens up vast array of possibilities, even in dimension 1. In particular, non-Hausdorff manifolds may have any cardinality from \(\mathfrak c\) upwards and 1 need no longer be orientable. Homogeneity is also lost: indeed, we exhibit 1-manifold whose only self-homeomorphism identity. A reasonable classification these seems infeasible Despite their esoteric nature, do appear naturally as leaf space foliated (Hausdorff) manifold. Even one-dimensional foliations plane resulting interesting use this connection to rigid foliation plane, i.e., with property that self-homeomorphisms respecting leaves map each itself. Non-Hausdorff possible models space-time ‘many-worlds’ interpretations quantum mechanics, relating time travel reduced twistor spaces relativity theory (see, example, [5], [11, pp. 594–595], [12, 249–255] [14]).