Sphere orderings representation of space-time

A space-time may be regarded as a partially ordered set by causal relations. R. J. Low considered a continuous sphere order representation of a space-time and obtained some interesting results. However, some important questions about the conditions of continuity of this representation are still left unanswered. In this paper, we considered the problem of continuity and presented a partial solution for the set of all discontinuity points of a sphere order representation and constructed an example of a space-time which could not admit a sphere order representation with a special dimension. We also showed that, for a poset, the definition of the 2-sphere ordering dimension is equivalent to the definition of the 2-Euclidian dimension.