The Attached Point Topology of the Abstract Boundary For Space-Time

Singularities play an important role in General Relativity and have been shown to be an inherent feature of most physically reasonable space-times. Despite this, there are many aspects of singularities that are not qualitatively or quantitatively understood. The abstract boundary construction of Scott and Szekeres has proven to be a flexible tool with which to study the singular points of a manifold. The abstract boundary construction provides a 'boundary' for any n-dimensional, paracompact, connected, Hausdorff, smooth manifold. Singularities may then be defined as entities in this boundary - the abstract boundary. In this paper a topology is defined, for the first time, for a manifold together with its abstract boundary. This topology, referred to as the attached point topology, thereby provides us with a description of how the abstract boundary is related to the underlying manifold. A number of interesting properties of the topology are considered, and in particular, it is demonstrated that the attached point topology is Hausdorff.