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

TL;DR

This paper introduces the attached point topology, a novel way to understand the relationship between a manifold and its abstract boundary, aiding the study of singularities in space-time within General Relativity.

Contribution

It defines the first topology for a manifold with its abstract boundary, providing new insights into the structure of singularities in space-time.

Findings

The attached point topology is Hausdorff.

The topology describes the relation between the manifold and its boundary.

Provides a new framework for studying singularities.

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.