Hausdorff separability of the boundaries for spacetimes and sequential spaces

TL;DR

This paper addresses the topological separation issues in spacetime boundaries in General Relativity, proposing a canonical refinement that ensures $T_2$-separability between spacetime and boundary points, applicable across various boundary types.

Contribution

It introduces a canonical minimal topology refinement that guarantees $T_2$-separability of spacetime and boundary points, resolving a longstanding topological issue in spacetime boundaries.

Findings

A canonical minimal refinement of the topology achieves $T_2$-separability.

The refined topology can be constructed from a modified limit operator $L^*$.

The approach applies to the causal boundary and allows separability to be an axiom.

Abstract

There are several ideal boundaries and completions in General Relativity sharing the topological property of being sequential, i.e., determined by the convergence of its sequences and, so, by some limit operator $L$. As emphasized in a classical article by Geroch, Liang and Wald, some of them have the property, commonly regarded as a drawback, that there are points of the spacetime $M$ non $T_1$-separated from points of the boundary $\partial M$. Here we show that this problem can be solved from a general topological viewpoint. In particular, there is a canonical minimum refinement of the topology in the completion $\overline{M}$ which $T_2$-separates the spacetime $M$ and its boundary $\partial M$ ---no matter the type of completion one chooses. Moreover, we analyze the case of sequential spaces and show how the refined $T_2$-separating topology can be constructed from a modification $L^*$ of the original limit operator $L$. Finally, we particularize this procedure to the case of the causal boundary and show how the separability of $M$ and $\partial M$ can be introduced as an abstract axiom in its definition.