The Endpoint Theorem

Abstract

The Endpoint Theorem links the existence of a sequence (curve), without accumulation points, in a manifold to the existence of an open embedding of that manifold so that the image of the given sequence (curve) has a unique endpoint. It plays a fundamental role in the theory of the Abstract Boundary as it implies that there is always an Abstract Boundary boundary point to represent the endpoint of such sequences and curves. The Endpoint Theorem will be of interest to researchers analysing specific spacetimes as it shows how to construct a chart in the original manifold which contains the sequence (curve). In particular, it has application to the study of singularities predicted by the singularity theorems.

arXiv commentIn memoriam of Chris J S Clarke. 12 pages, 1 figure. PACS: 04.20.Dw. Appeared in CQG: https://iopscience.iop.org/article/10.1088/1361-6382/abe1fe/meta

References in corpus (10)

• Singularity Theorems and Their Consequences
• The 1965 Penrose singularity theorem
• Causal Relationship: a new tool for the causal characterization of Lorentzian manifolds
• The abstract boundary---a new approach to singularities of manifolds
• Null distance on a spacetime
• An exact smooth Gowdy-symmetric generalized Taub-NUT solution
• Generalizations of the Abstract Boundary singularity theorem
• The Attached Point Topology of the Abstract Boundary For Space-Time
• A rigidity result on the ideal boundary structure of smooth space-times
• The chart based approach to studying the global structure of a spacetime induces a coordinate invariant boundary

Cited by in corpus (2)

• On the Definition of Black Holes: Bridging the Gap Between Black Holes and Singularities
• What actually happens when you approach a gravitational singularity?