The Endpoint Theorem

TL;DR

The Endpoint Theorem establishes a fundamental link between sequences without accumulation points in manifolds and the existence of open embeddings that reveal their endpoints, aiding in the analysis of spacetime singularities.

Contribution

It introduces the Endpoint Theorem, connecting sequence endpoints with open embeddings, advancing the understanding of boundary points in manifold theory.

Findings

Ensures existence of boundary points for sequences without accumulation points.

Provides a method to construct charts containing specific sequences.

Facilitates analysis of singularities in spacetime models.

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.