Hausdorff closed limits and the c-boundary I: a new topology for the c-completion of spacetimes

This is the first of two papers (see also Costa e Silva et al (2019)) in which we argue for the benefits of considering the closed limit topology (CLT) on the Geroch–Kronheimer–Penrose causal completion of globally hyperbolic spacetimes with (possibly empty) smooth timelike boundary  . We show here that this topology, introduced by Hausdorff himself (see Hausdorff, is not only metrizable, but retains essentially all the good properties of certain topologies previously defined in this ambient. In particular, 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. (paper)