On a Possible Continuous Analogue of the Szpilrajn Theorem and its Strengthening by Dushnik and Miller

The Szpilrajn theorem and its strengthening by Dushnik and Miller belong to the most quoted theorems in many fields of pure and applied mathematics as, for instance, order theory, mathematical logic, computer sciences, mathematical social sciences, mathematical economics, computability theory and fuzzy mathematics. The Szpilrajn theorem states that every partial order can be refined or extended to a total (linear) order. The theorem by Dushnik and Miller states, moreover, that every partial order is the intersection of its total (linear) refinements or extensions. Since in mathematical social sciences or, more general, in any theory that combines the concepts of topology and order one is mainly interested in continuous total orders or preorders in this paper some aspects of a possible continuous analogue of the Szpilrajn theorem and its strengthening by Dushnik and Miller will be discussed. In particular, necessary and sufficient conditions for a topological space to satisfy a possible continuous analogue of the Dushnik-Miller theorem will be presented. In addition, it will be proved that a continuous analogue of the Szpilrajn theorem does not hold in general. Further, necessary and in some cases necessary and sufficient conditions for a topological space to satisfy a possible continuous analogue of the Szpilrajn theorem will be presented.