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

Studia Logica 100 (1-2):419-435 (2012)
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 $${\preccurlyeq}$$ can be uniquely reconstructed if we know the “interior” $${\prec}$$ of the order relation. It is also known that in some cases, we can uniquely reconstruct $${\prec}$$ (and hence, topology) from $${\preccurlyeq}$$. In this paper, we show that, in general, under reasonable conditions, the open order $${\prec}$$ (and hence, the corresponding topology) can be uniquely determined from its closure $${\preccurlyeq}$$
Keywords ordered topological space  order-preserving mappings  open and closed orders  space-time geometry  logic
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DOI 10.1007/s11225-012-9386-y
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E. C. Zeeman (1963). Causality Implies the Lorentz Group. Journal of Mathematical Physics 5 (4):490-493.
John L. Kelley (1962). General Topology. Journal of Symbolic Logic 27 (2):235-235.

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