Topological Ordered Spaces
In this chapter we examine the symbiotic relationship existing between topologies and partial orders. We establish theories of completeness, compactness, and normality that include the usual uniform and topological theories in the special case that the partial order under consideration is equality. Although quasi-uniformities do not make their appearance until the third section of this chapter, their role is central to the study of the interdependence between topologies and orders. Indeed it is the theory of quasi-uniformities that enables us to consider ordered completions and compactifications of ordered spaces and to develop for generalized ordered spaces a theory of uniformities that reflects both the topological and order structures of these spaces.