Hicks in his book, "Notes on differential geometry", works with manifolds by specifically stating that they are not required to be Hausdorff unless otherwise stated. He then goes on to define the usual manifold structures, vector fields, connections, metric tensors, etc. In chapter 6, he defines the usual length of a curve by a Riemannian metric tensor, that is, for a curve $\gamma$ with tangent vector $T$, the length between the point $\gamma(p)$ and $\gamma(q)$ is

$$|\gamma|^q_p = \int_p^q \sqrt{\langle T(\lambda), T(\lambda)\rangle} d\lambda$$

which he then uses for the definition of a distance function

$$d(p,q) = \inf [|\gamma| , \gamma \text{ piecewise continuous between $p$ and $q$}]$$

which can be shown to be a pseudometric on the manifold, which specializes to a metric if the manifold is Hausdorff. The problem is, a pseudometric space cannot be $T_0$ and not also a metric space, and every manifold is itself a $T_0$ space. And by the Smirnov metrization theorem, a non-Hausdorff space cannot be a metric space.

I'm not sure exactly where the problem comes from here. Is there a problem of existence of one of the structures involved? Is the metric tensor not continuous in those cases (that is one of the assumption of the proof)? More generally, do such manifolds admit global metric sections for arbitrary signatures, in a different way than Hausdorff manifolds, that is?

differentfrom the topology as a non-Hausdorff manifold. $\endgroup$ – Chris Schommer-Pries Apr 21 '17 at 19:02