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

TL;DR

This paper introduces the strongly attached point topology for the abstract boundary of manifolds, enhancing the connection between the boundary and the manifold, and providing a more natural topological framework for analyzing singularities.

Contribution

The paper presents a new strongly attached point topology that improves the connectivity between a manifold and its abstract boundary compared to previous topologies.

Findings

The strongly attached point topology connects the boundary more closely to the manifold.

It exhibits properties supporting its naturalness and appropriateness for boundary analysis.

The topology enhances understanding of singularities within the boundary framework.

Abstract

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.