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We discuss the topological nature of the boundary spacetime, the conformalinfinity of the ambient cosmological metric. Due to the existence of ahomothetic group, the bounding spacetime must be equipped not with the usualEuclidean metric topology but with the Zeeman fine topology. This then placessevere constraints to the convergence of a sequence of causal curves and theextraction of a limit curve, and also to our ability to formulate conditionsfor singularity formation.

Topology of the ambient boundary and the convergence of causal curves