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I would like to know how the standard differential geometry of manifolds would change if we didn't assume the Hausdorff condition and/or the second axiom of countability. There are some simple things that can be said at a first glance:

  • Manifolds would not necessarily admit partitions of unity, so not every Manifold would admit a Riemannian metric.

  • A Haussdorf manifold admitting a connection would be automatically second-countable.

  • A Haussdorf manifold admitting a Lorentzian metric would be automatically paracompact.

  • Manifolds without these two assumptions wouldn't necessarily have the homotopy type of a CW-complex.

Since the question is maybe too broad, we can focus for example on the following especific issue:

How Berger's classification of manifolds of special holonomy would change without these two assumptions?

One motivation to ask this question is that manifolds describing space-times needn't to be Haussdorf, or at least physicists have studied non-Haussdorf solutions to General Relativity, see for example "The large scale structure of spacetime" of Hawking and Ellis.

Thanks.

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  • $\begingroup$ “A Haussdorf manifold admitting a connection would be automatically second-countable.”: This is false, an uncountable disjoint union of R^n is Hausdorff, paracompact, and admits a connection on its tangent bundle, yet it is not second countable. $\endgroup$
    – Dmitri Pavlov
    Commented Aug 23, 2015 at 16:10

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Probably the simplest example of a non-second-countable manifold is the long line, and it already exhibits some interesting phenomena. See for example "Various smoothings of the long line and their tangent bundles" by Peter Nyikos [Advances in Math. 93 (1992) 129-213].

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    $\begingroup$ I know that there are known relatively many non-second-countable manifolds. That was not the question. The question was more oriented towards how important results in differential geometry, such as Berger's classification of manifolds with special holonomy should be modified in this more relaxed setting. $\endgroup$
    – Bilateral
    Commented Aug 22, 2015 at 21:01

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