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

The abstract boundary construction of Scott and Szekeres provides a `boundary for any n-dimensional, paracompact, connected, Hausdorff, smooth manifold. Singularities may then be defined as objects within this boundary. In a previous paper by the authors, a topology referred to as the attached point topology was defined for a manifold and its abstract boundary, thereby providing us with a description of how the abstract boundary is related to the underlying manifold. In this paper, a second topology, referred to as the strongly attached point topology, is presented for the abstract boundary construction. Whereas the abstract boundary was effectively disconnected from the manifold in the attached point topology, it is very much connected in the strongly attached point topology. A number of other interesting properties of the strongly attached point topology are considered, each of which support the idea that it is a very natural and appropriate topology for a manifold and its abstract boundary.

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.