There are two notions of "metric" and "pseudometric" here: As
tensor fields on a manifold, and as distance functions. I always
mean the former (which I would regard as the more interesting).
Let's accept that definition. If I were doing this, I would
try to find an example of a (always, paracompact) (non-Hausdorff)
manifold that does not admit any (smooth) Riemannian metric.
My guess is that the non-Hausdorff manifolds are divided into
two classes: the well-behaved ones (that admit Riemannian and