Preorderable topologies and order-representability of topological spaces

Campión, María Jesús; Candeal, Juan Carlos; Induráin, Esteban

doi:10.1016/j.topol.2009.01.018

“…In addition, τ is said to be lower preorderable if there exists a total preorder on X such that τ coincides with the right ≺-topology induced by the asymmetric binary relation ≺ on X . (See also [9]). …”

Section: Definition 27mentioning

confidence: 98%

“…Remark 2.6 About the notation and nomenclature used in Definition 2.5, we point out that in some papers (see e.g., [9,14,17]) the left R-topology (respectively, the right R-topology) is called "upper topology" (respectively, "lower topology"). However, that nomenclature may cause confusion when compared to the traditional one in the domain theory (see [15]), where, if a partial order is fixed on a set X , then usually the upper topology is the least topology such that all the principal lower sets {y ∈ X : y x}, x ∈ X , are closed.…”

Section: Definition 24mentioning

confidence: 99%

“…Classical results concerning different kinds of orderability of topologies defined on a set X can be seen in [9,20,22,24,34]. In particular, in [24] the topologies for which there exists a linear order on X whose associated asymmetric relation ≺ induces the topology, namely the so-called orderable topologies, were already characterized.…”

Section: Remark 28mentioning

confidence: 99%

“…Basically, an ordinal family consists of a suitable nested collection of subsets of a given nonempty set where a topology has been defined. Through these families we may induce semicontinuous (respectively, continuous) total preorders on the given set and analyze their numerical representability by means of semicontinuous (respectively, continuous) real-valued order-preserving functions (see also [4,8]). …”

mentioning

confidence: 99%

“…Using also these kinds of families, in [9] the topological spaces whose topology is induced by a total preorder (namely, the so-called preorderable topologies) were also characterized.…”

mentioning

confidence: 99%

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Induráin

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We study binary relations and topologies induced by means of nested collections of subsets of a given nonempty set. Dually, given a topological space, we characterize properties of the topology in terms of suitable nested families of open subsets. By means of this relationship between nested collections of sets and nested topologies, we study the representability of total preorders via (semi-)continuous realvalued order isomorphisms.Keywords Subbasis of a topology · Nested families of subsets of a set · Topologies induced by binary relations · Nested topologies · Ordinal representability properties of topologies Mathematics Subject Classification Primary: 54A05 · Secondary: 54A10 · 54F05

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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

¹

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Zuanon

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Mathematical Topics on Representations of Ordered Structures and Utility Theory

View full text Add to dashboard Cite

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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Preorderable topologies and order-representability of topological spaces

Cited by 21 publications

References 15 publications

Topologies Generated by Nested Collections

Topologies Generated by Nested Collections

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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