Non-Hausdorff Manifolds and Foliations

Abstract

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