Invariance properties of boundary sets of open embeddings of manifolds and their application to the abstract boundary

TL;DR

This paper investigates the topological invariance of boundary sets in manifolds' abstract boundaries, with implications for understanding singularities and infinity points in general relativity.

Contribution

It introduces invariance properties of boundary sets, including compactness and isolation, within the abstract boundary framework, applicable to various geometric contexts.

Findings

Compactness is invariant under boundary set equivalence.

Introduces the concept of boundary set isolation.

Applicable to manifolds in general relativity.

Abstract

The {\em abstract boundary\/} (or {\em {\em a\/}-boundary\/}) of Scott and Szekeres \cite{Scott94} constitutes a ``boundary'' to any $n$-dimensional, paracompact, connected, Hausdorff, $C^\infty$-manifold (without a boundary in the usual sense). In general relativity one deals with a {\em space-% time\/} $(\cM,g)$ (a 4-dimensional manifold $\cM$ with a Lorentzian metric $g$), together with a chosen preferred class of curves in $\cM$. In this case the {\em a\/}-boundary points may represent ``singularities'' or ``points at infinity''. Since the {\em a\/}-boundary itself, however, does not depend on the existence of further structure on the manifold such as a Lorentzian metric or connection, it is possible for it to be used in many contexts. In this paper we develop some purely topological properties of abstract boundary sets and abstract boundary points ({\em a\/}-boundary points). We prove, amongst other things, that compactness is invariant under boundary set equivalence, and introduce another invariant concept ({\em isolation\/}), which encapsulates the notion that a boundary set is ``separated'' from other boundary points of the same embedding. ....... [The abstract continues in paper proper - truncated to fit here.]