We present a one-to-one correspondence between equivalence classes of
embeddings of a manifold (into a larger manifold of the same dimension) and
equivalence classes of certain distances on the manifold. This correspondence
allows us to use the Abstract Boundary to describe the structure of the `edge'
of our manifold without resorting to structures external to the manifold
itself. This is particularly important in the study of singularities within
General Relativity where singularities lie on this `edge'. The ability to talk
about the same objects, e.g., singularities, via different structures provides
alternative routes for investigation which can be invaluable in the pursuit of
physically motivated problems where certain types of information are
unavailable or difficult to use.Comment: 23 page
