Reconstructing an Open Order from Its Closure, with Applications to Space-Time Physics and to Logic

In his logical papers, Leo Esakia studied corresponding ordered topological spaces and order-preserving mappings. Similar spaces and mappings appear in many other application areas such the analysis of causality in space-time. It is known that under reasonable conditions, both the topology and the original order relation ≤ can be uniquely reconstructed if we know the interior < of the order relation. It is also known that in some cases, we can uniquely reconstruct < (and hence, topology) from ≤. In this paper, we show that, in general, under reasonable conditions, the open order < (and hence, the corresponding topology) can be uniquely determined from its closure ≤.