Abstract
Classically the order topology on a linearly ordered set is the topology generated by the "open rays". This dissertation is concerned with a three fold generalization of this situation: The category of Sets is replaced by an arbitrary category B with appropriate properties, linear order is replaced by a relation, and the "topology" is replaced by a topological category. The main theoretical problem here is to reformulate the concept of an ordered topological space in terms available in a general topological category; a situation in which the notions of points, open and closed sets, and rays may be meaningless. Our solution to this problem is based on the observation that the order topology on the standard unit interval renders the product topology on the standard unit n-cube coherent with respect to the family of order defined n-simplexes (n! in number) making up the cube. This coherence condition, appropriately modified, is the bases of our concept of an n-ordered topological object with respect to a relation. Besides developing the theoretical foundations necessary to rigorously formulate the "n-ordered topological object" concept, many explicit characterizations of 2-ordered and n-ordered topological objects are given for various topological categories of classical interest. One of the more important cases is that of topological spaces in which the 2-ordered topological structures for reflexive, antisymmetric relations are characterized in terms of 2-ply covers. This characterization reduces to the classical notion for Hausdorff topological spaces but includes the non-Hausdorff cases as well; useful in the algebraic topological study of finite topological spaces, for example, in which Hausdorff intervals are all discrete.Several open problems and conjectures are also discussed.