Some necessary and sufficient conditions for admitting a continuous sphere order representation of two-dimensional space-times

A space-time may be regarded as a partially ordered set by causal relations. Recently, as a poset, a sphere order representation of a space-time is considered and some interesting results are obtained. In this paper, we define a weakly causally convex open set and characterize connected open subsets of the n-dimensional Minkowski space-time which admit a continuous n-sphere order representation. Then, some necessary and sufficient conditions for admitting a continuous two-sphere order representation of two-dimensional space-times, are given. As a result, it is shown that any simply connected and causally simple space-time of dimension two can be causally isomorphically imbedded into the two-dimensional Minkowski space-time.