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

Content Provider	Springer Nature Link
Author	Herden, Gerhard Bosi, Gianni
Copyright Year	2007
Abstract	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.
Ending Page	296
Page Count	26
Starting Page	271
File Format	PDF
ISSN	01678094
e-ISSN	15729273
Journal	Order
Issue Number	4
Volume Number	23
Language	English
Publisher	Kluwer Academic Publishers
Publisher Date	2007-01-16
Publisher Place	Dordrecht
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Geometry κ-attracting set Szpilrajn property Utility theory Total order Convex and Discrete Geometry Theory of Computation Linearly ordered topological spaces, generalized ordered spaces, and partially ordered spaces weakly continuous binary relation
Content Type	Text
Resource Type	Article
Subject	Algebra and Number Theory Computational Theory and Mathematics Geometry and Topology

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