Non-Hausdorff Manifolds and Foliations

David Gauld ()
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David Gauld: University of Auckland, Department of Mathematics

Chapter Chapter 9 in Non-metrisable Manifolds, 2014, pp 153-166 from Springer

Abstract: 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 $$\mathfrak c$$ 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]).

Keywords: Twistor Space; Order Topology; Connected Space; Leaf Space; Disjoint Neighbourhood (search for similar items in EconPapers)
Date: 2014
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DOI: 10.1007/978-981-287-257-9_9

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