Abstract
This thesis is written within the framework of the abstract boundary (or aboundary) of Scott and Szekeres, [24]. The a-boundary provides a concept of “boundary” for any n-dimensional, paracompact, connected, Hausdorff manifold, defined in such a way that the boundary is independant of the particular embedding used to display the manifold. This makes it possible to define various types of boundary points of space-time such as “singularities” and “points at infinity”. The original research that will be presented in this thesis can be roughly divided up into two categories; results relating to the existence of optimal embeddings of solutions to Einstein’s Field Equations and a-boundary singularity theorems. In addition, the implications of the “finite connected neighbourhood region property” and the bounded “acceleration” property are explored. It is also shown that not all space-times are maximally extendable.
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