Metrisability of Manifolds
by
David Gauld
The University of Auckland

In this talk by a manifold is meant a topological space which is connected, Hausdorff and locally euclidean.

Over the past century a considerable amount of effort has gone into the study of small (=compact or \sigma-compact) manifolds. More recently quite a bit of effort has gone into the study of larger manifolds. At the same time there has been a lot of study of the question of when a topological space is metrisable. It turns out that metrisability forms an interesting border between small manifolds and large.

In my talk I shall discuss some of the many topological conditions which I have collected and which are equivalent to metrisability for a manifold but not for a topological space in general. Not surprisingly many conditions are weaker than metrisability in general: the conditions of being a manifold are sufficient to promote a weak condition to metrisability. Not surprisingly for the same reason many conditions are stronger than metrisability in general. Again not surprisingly there are conditions equivalent to metrisability for a manifold which are unrelated for a general space. Perhaps also not surprisingly quite a few authors list the demand that their manifolds satisfy a list of properties each of which is equivalent to metrisability. The list also has some notable omissions. Some of the conditions are somewhat geometric, many involve covering properties, and some seem rather odd at first sight.

Date received: June 5, 2001