This talk was in two parts, one considering conditions related to and even equivalent to metrisability for a manifold and the other a description of an application combining Algebraic Topology and Set Theory to the solution of a problem in topology.

Not surprisingly there are many conditions which are equivalent to metrisability for a topological manifold but not for a general topological space; 45 are listed here. Some of these conditions are strictly weaker, some strictly stronger and others unrelated to metrisability in a general topological space. There is also a discussion of some other conditions which have been introduced recently and which are strictly weaker than metrisability for a manifold.

The main tool from outside topology which is used to study large spaces in general and non-metrisable manifolds in particular is Set Theory. On the other hand Algebra has been extremely successful as a tool in the study of compact manifolds. Recently some techniques of Algebraic Topology have been combined with ideas from Set Theory to determine the torsion of the group of homeomorphisms of powers of the long line.

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