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

Bosi, Gianni; Herden, Gerhard

doi:10.1007/s11083-006-9047-8

Cited by 33 publications

(42 citation statements)

References 45 publications

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“…In addition, we choose for every ordinal number < the subset I :¼ f 2 À j g of À. It follows that b :¼ ffg j 2 À n fgg [ fI j 2 À n fgg is a basis of a topology t on À. Theorem 2.1 implies with the help of Remark 2.2 that ðÀ; t Þ satisfies a strong and a continuous analogue of the Szpilrajn theorem (see also [3,Proposition 4…”

Section: For Every Pointmentioning

confidence: 99%

“…This result follows by a straightforward modification of the proof of Theorem 2.1. Since this modification at least partly is presented in the proof of Proposition 4.4 in Bosi and Herden [3] we may omit the details for the sake of brevity. EXAMPLE 2.3.…”

Section: For Every Pointmentioning

confidence: 99%

“…Bosi and Herden [3,Proposition 4.3]), in contrast to the usual notation in the literature, we do not speak of an extension but of a refinement of " by a total order e.…”

Section: Introductionmentioning

confidence: 99%

“…Therefore, in Herden and Pallack [12] and Bosi and Herden [3] a condition has been presented that necessarily must be satisfied by every order on a topological space that possibly has a continuous total refinement or possibly is the intersection of its continuous total refinements. Orders on a topological space that satisfy this necessary condition are said to be weakly continuous.…”

Section: Introductionmentioning

confidence: 99%

“…Therefore, continuous analogues of the Szpilrajn theorem and the Dushnik-Miller theorem respectively that consider arbitrary weakly continuous preorders are studied by the authors in a separate paper (cf. [3]). Nevertheless, topologies t on X that satisfy a strong continuous analogue of the Szpilrajn theorem and of the Dushnik-Miller theorem respectively also satisfy a continuous analogue of the Szpilrajn theorem and the Dushnik-Miller theorem respectively (cf.…”

Section: Introductionmentioning

confidence: 99%

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

Bosi

Herden

2005

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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. (2000): 54F05, 91B16, 06A05. Mathematics Subject Classification

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Section: For Every Pointmentioning

confidence: 99%

Section: For Every Pointmentioning

confidence: 99%

“…Bosi and Herden [3,Proposition 4.3]), in contrast to the usual notation in the literature, we do not speak of an extension but of a refinement of " by a total order e.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Bosi

Herden

2005

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Mathematical Topics on Representations of Ordered Structures and Utility Theory

2020

Studies in Systems, Decision and Control

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A continuous multi-utility fully represents a not necessarily total preorder on a topological space by means of a family of continuous increasing functions. While it is very attractive for obvious reasons, and therefore it has been applied in different contexts, such as expected utility for example, it is nevertheless very restrictive.In this paper we first present some general characterizations of the existence of a continuous order-preserving function, and respectively a continuous multi-utility representation, for a preorder on a topological space. We then illustrate the restrictiveness associated to the existence of a continuous multi-utility representation, by referring both to appropriate continuity conditions which must be satisfied by a preorder admitting this kind of representation, and to the Hausdorff property of the quotient order topology corresponding to the equivalence relation induced by the preorder. We prove a very restrictive result, which may concisely described as follows: the continuous multi-utility representability of all closed (or equivalently weakly continuous) preorders on a topological space is equivalent to the requirement according to which the quotient topology with respect to the equivalence corresponding to the coincidence of all continuous functions is discrete.

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Continuity and Continuous Multi-utility Representations of Nontotal Preorders: Some Considerations Concerning Restrictiveness

Bosi

Zuanon

2020

Mathematical Topics on Representations of Ordered Structures and Utility Theory

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

Abstract: These data suggest that facial emotion processing can be impaired in MCI prior to the more marked cognitive deficits seen with clinically diagnosed Alzheimer disease.

Cited by 33 publications

References 45 publications

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

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

Mathematical Topics on Representations of Ordered Structures and Utility Theory

Continuity and Continuous Multi-utility Representations of Nontotal Preorders: Some Considerations Concerning Restrictiveness

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