Topological spaces for which every continuous total preorder can be represented by a continuous utility function

Herden, Gerhard

doi:10.1016/0165-4896(91)90002-9

Cited by 33 publications

(23 citation statements)

References 7 publications

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“…In Herden [11] a topology t on X is said to be useful if every continuous total preorder on X has a continuous utility representation. In this sense the theorems of Eilenberg and Debreu present sufficient conditions for a topology t on X to be useful.…”

Section: Debreumentioning

confidence: 99%

Continuous Utility Representation Theorems in Arbitrary Concrete Categories

Bosi

Herden

2007

Appl Categor Struct

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In this paper the continuous utility representation problem will be discussed in arbitrary concrete categories. In particular, generalizations of the utility representation theorems of Eilenberg, Debreu and Estévez and Hervés will be presented that also hold if the codomain of a utility function is an arbitrary totally ordered set and not just the real line. In addition, we shall prove and apply a general result on the characterization of structures that have the property that every continuous total preorder has a continuous utility representation. Finally, generalizations of the utility representation theorems of Debreu and Eilenberg will be discussed that are valid if we consider arbitrary binary relations and allow a utility function to have values in an arbitrary totally ordered set.

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Section: Debreumentioning

confidence: 99%

Continuous Utility Representation Theorems in Arbitrary Concrete Categories

Bosi

Herden

2007

Appl Categor Struct

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“…Given a nonempty set X endowed with a topology τ (i.e., (X, τ ) is a topological space), the topology τ on X is said to have the continuous representability property (CRP) if every continuous total preorder ≾ defined on X admits a representation by means of a continuous order-monomorphism. Topologies of this kind were introduced by Herden [25] under the name of "useful topologies" (see also Herden and Pallack [26]). Among the topologies that have CRP are the second countable ones (see Debreu [17]), the connected plus separable ones (see Eilenberg [19]) and the locally connected plus separable ones (see Campión et al [8]).…”

Section: Preliminariesmentioning

confidence: 99%

Continuous Order Representability Properties of Topological Spaces and Algebraic Structures

Campión¹,

Candeal²,

Induráin³

et al. 2012

Journal of the Korean Mathematical Society

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“…Herden [9] introduced the concept of a useful topology as a topology τ on an arbitrary set X such that every continuous total preorder on the topological space (X, τ ) is representable by means of a continuous utility function (see also Herden and Pallack [10]).…”

Section: Introductionmentioning

confidence: 99%

Semicontinuous representability of interval orders on a metrizable topological space

Bosi¹,

Zuanon²

2007

ijcms

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An interval order on an arbitrary set X is said to be representable by a pair (u, v) of real-valued functions on X if x y is equivalent to u(x) ≤ v(y) for all x, y ∈ X. We say that a topology τ on X is completely useful in the sense of an interval order (i.o. completely useful for the sake of brevity) if every upper semicontinuous interval order on the topological space (X, τ) is representable by a pair (u, v) of real-valued functions with v upper semicontinuous. We show that every i.o. completely useful topology on X is also completely useful (i.e., every upper semicontinuous total preorder on (X, τ) is representable by an upper semicontinuous utility function). Further we show that a metrizable topology τ on X is i.o. completely useful if and only if τ is separable.

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Topological spaces for which every continuous total preorder can be represented by a continuous utility function

Cited by 33 publications

References 7 publications

Continuous Utility Representation Theorems in Arbitrary Concrete Categories

Continuous Utility Representation Theorems in Arbitrary Concrete Categories

Continuous Order Representability Properties of Topological Spaces and Algebraic Structures

Semicontinuous representability of interval orders on a metrizable topological space

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