A topological preordered space admits a Hausdorff T2-preorder compactification if and only if it is Tychonoff and the preorder is represented by the family of continuous isotone functions. We construct the largest Hausdorff T2-preorder compactification for these spaces and clarify its relation with Nachbin's compactification. Under local compactness the problem of the existence and identification of the smallest Hausdorff T2-preorder compactification is considered.2010 MSC: 54E15 (primary), 54F05, 54E55, 06F30 (secondary).

We consider the problem of determining from intrinsic properties whether or not a given spacelike surface is a Cauchy surface. We present three results relevant to this question. First, we derive necessary and sufficient conditions for a compact surface to be a Cauchy surface in a spacetime which admits one. Second, we show that for a non-compact surface it is impossible to determine whether or not it is a Cauchy surface without imposing some restriction on the entire spacetime. Third, we derive conditions for an asymptotically flat surface to be a Cauchy surface by imposing the global condition that it be imbedded in a weakly asymptotically simple and empty spacetime.

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Several properties of instantaneous Cauchy surfaces are obtained. It is shown that a strongly causal spacetime admits an instantaneous Cauchy surface through each of its points, that there is a close and reversible relationship between these surfaces and maximal open globally hyperbolic subsets, that every instantaneous Cauchy surface is contained in a maximal instantaneous Cauchy surface, and that the latter surface is a maximal achronal surface which separates spacetime into past, present, and future. Some other properties of instantaneous Cauchy surfaces are discussed along with a refinement of an earlier topology change property.