  The problem of classifying boundary points of space-time, for example
singularities, regular points and points at infinity, is an unexpectedly subtle
one. Due to the fact that whether or not two boundary points are identified or
even "nearby" is dependant on the way the space-time is embedded, difficulties
occur when singularities are thought of as an inherently local aspect of a
space-time, as an analogy with electromagnetism would imply. The completion of
a manifold with respect to a pseudo-Riemannian metric can be defined
intrinsically, [SS94]. This is done via an equivalence relation, formalising
which boundary sets cover other sets. This paper works through the
possibilities, providing examples to show that all covering relations not
immediately ruled out by the definitions are possible.

General Relativity and Quantum Cosmology