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arXiv · 1408.4201

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

Richard A. Barry·Susan M. Scott
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Abstract

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.

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Richard A. Barry, Susan M. Scott. 2014-08-19. The Attached Point Topology of the Abstract Boundary For Space-Time. https://doi.org/10.1088/0264-9381%2F28%2F16%2F165003

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