Generalized Scott Topology on Sets with Families of Pre-orders

[Fan L., W. Ji and W. L. Wang. “The Information Order Approximation and Generalized Chains' Completion”, Beijing: Capital Normal University, Preprint, 2005. (in Chinese), Lei Fan] proposed a class of sets with families of pre-orders (R-posets for short). They are not only a non-symmetric generalization of sfe [Monteiro L., Semantic Domains Based on Sets with Families of Equivalences. Electronic Notes in Theoretical Computer Science 11 (1998): 1–34, L.Monteiro] but also a special case of quasi-metric spaces (qms, [Smyth M. B., Quasi Uniformities: Reconciling Domains with Metric Spaces. Proceedings of the 3rd Workshop on Mathematical Foundations of Programming Language Semantics, APR. 8-10, 1987. Lecture Notes In Computer Science, Vol. 298 (1988): 236-253. Springer-Verlag, Berlin, M. B. Smyth]) and generalized ultrametric spaces (gums, [Rutten J. J. M. M., “Elements of Generalized Ultrametric Domain Theory”. Technical Report CS-R9507, CWI, Amsterdam, 1995, J. J. M. M. Rutten]). In this paper, we define a kind of generalized Scott topology on R-posets and discuss some basic properties of the topology. Some relevant interesting examples are offered. It is worth pointing out that an R-monotone functions is R-continuous if and only if (iff for short) it's continuous with respect to (w.r.t for short) the generalized Scott topology.