Bicontinuous Domains and Some Old Problems in Domain Theory

K. Martin and P. Panangaden [Martin, K., and P. Panangaden, A domain of space-time intervals in general relativity, Communications in Mathematical Physics, 267 (2006), pages 563–586] have made the interesting observation that the causal order in general relativity theory closely relates to domain theoretical notions. From the causal structure of models of space-time which are called strongly hyperbolic they extract the notion of a strongly hyperbolic poset. These are in particular bicontinuous, that is, continuous posets with the property that they are also continuous with respect to the opposite order. Not much is known about bicontinuous posets. In this paper we collect known results, we exhibit substantial examples, we show that these examples are connected with open problems in domain theory, and we indicate directions of research.