Ths is another step to improve the Geroch-Kronheimer-Penrose boundary construction for space-times. A new identification rule is proposed on the set of ideal points of a space-time which seems to be more satisfactory in the stably causal case. But it remains difficult to describe the corresponding equivalence relation explicitly. Also the author does not try to construct a Hausdorff topology on the extended space by requiring further identifications. The topological requirements are restricted to the property that causal curves have a unique end point. This can be guaranteed in special cases.