Hausdorff closed limits and the causal boundary of globally hyperbolic spacetimes with timelike boundary

We show that when a spacetime $\mathcal{M}(=M \cup \partial M)$ is globally hyperbolic with (possibly empty) smooth timelike boundary $\partial M$, a metrizable topology, the closed limit topology (CLT) introduced by F. Hausdorff himself in the 1950's in set theory, can be advantageously adopted on the Geroch-Kronheimer-Penrose causal completion of M, retaining essentially all the good properties of previous topologies in this ambient. In particular, we show that if the globally hyperbolic spacetime $M$ admits a conformal boundary, defined in such broad terms as to include all the standard examples in the literature, then the latter is homeomorphic to the causal boundary endowed with the CLT. We also discuss how our recent proposal arXiv:1807.00152 for a definition of null infinity using only causal boundaries can be translated when using the CLT, simplifying a number of technical aspects in the pertinent definitions. In a more technical vein, in the appendix we discuss the relationship of the CLT with the more generally applicable (but not always Hausdorff) chronological topology, and show that they coincide exactly in those cases when the latter happens to be Hausdorff.