The causal structure of spacetime defines a (pair of) natural order structures on the underlying set of events. Much of the analysis of cauasl structure involves a delicate interplay between order, topology and geometry. In view of the fundamental role of the causal order in certain approaches to quantum gravity as well as its fundamental role in concurrency theory one can ask whether the topology can be derived from pure order theoretic considerations.

In a remarkable example of serendipity, order theory has been developed by computer scientists and mathematicians in order to capture computability concepts. Dana Scott developed a notion of a continuous lattice or continuous poset with a view to capturing computability as continuity with a suitable topology that has come to be known as the Scott topology. This subject has acquired the name of ``domain theory.''

We applied domain theory to the problem of reconstructing the spacetime topology from the order and came up with a number of results about reconstruction of spacetime structure from just a countable dense set.

We prove that a globally hyperbolic spacetime with its causality relation is a bicontinuous poset whose interval topology is the manifold topology. From this one can show that from only a countable dense set of events and the causality relation, it is possible to reconstruct a globally hyperbolic spacetime in a purely order theoretic manner. The ultimate reason for this is that globally hyperbolic spacetimes belong to a category that is equivalent to a special category of domains called interval domains.

This was joint work with Keye Martin of Naval Research Laboratories.