Abstract
The Szpilrajn theorem states that every (partial) order has a total (linear) refinement or extension by a total (linear) order. Its strengthening by Dushnik and Miller states, moreover, that every (partial) order is the intersection of its total (linear) refinements or extensions. In any theory that combines the concepts of topology and order, however, one is interested in (weakly) continuous orders. Therefore, in this paper the question will be answered, if the Szpilrajn theorem and its strengthening by Dushnik and Miller respectively can be generalized in such a way that they also include the case that (weakly) continuous orders are considered. Since arbitrary preorders are not considered in this paper we speak of possible strong continuous analogues of the Szpilrajn theorem and its strengthening by Dushnik and Miller. The main results of this paper show that the Dushnik–Miller theorem cannot be generalized to the (weakly) continuous case while the Szpilrajn theorem at least in very particular situations allows generalizations to the case that (weakly) continuous orders are considered. Finally, also a strong semicontinuous analogue of the Szpilrajn theorem and its strengthening by Dushnik and Miller will be discussed.
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Bosi, G., Herden, G. On a Strong Continuous Analogue of the Szpilrajn Theorem and its Strengthening by Dushnik and Miller. Order 22, 329–342 (2005). https://doi.org/10.1007/s11083-005-9022-9
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DOI: https://doi.org/10.1007/s11083-005-9022-9