A topological characterization of the existence of non-empty choice sets

Andrikopoulos, Athanasios; Zacharias, Eleftherios

doi:10.1016/j.topol.2011.04.030

Cited by 4 publications

(2 citation statements)

References 19 publications

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“…An implicit 1 presence of that same point can be found in the more recent Bosi and Zuanon [8,Corollary 3.2] and Quartieri [22,Theorem 4], where the authors provide, respectively, characterizations of the existence of unconstrained maximals of a preorder and of the existence of maximals of an arbitrary relation on every nonempty compact subset of its ground set. Somehow relatedly, the same point is tacitly present also in Andrikopoulos and Zacharias [5], where Smith and Schwartz sets are considered with respect to fixed choice sets. None of the results mentioned so far provides purely topological conditions that characterize the existence of optimal points on every nonempty compact subset of the ground set of an objective relation (or, in the case of functions, of the domain of an objective function).…”

Section: Introductionmentioning

confidence: 75%

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On the Existence of Greatest Elements and Maximizers

Quartieri

2022

J Optim Theory Appl

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We obtain several characterizations of the existence of greatest elements of a total preorder. The characterizations pertain to the existence of unconstrained greatest elements of a total preorder and to the existence of constrained greatest elements of a total preorder on every nonempty compact subset of its ground set. The necessary and sufficient conditions are purely topological and, in the case of constrained greatest elements, are formulated by making use of a preorder relation on the set of all topologies that can be defined on the ground set of the objective relation. Observing that every function into a totally ordered set can be naturally conceived as a total preorder, we then reformulate the mentioned characterizations in the more restrictive case of an objective function with a totally ordered codomain. The reformulations are expressed in terms of upper semi- and pseudo-continuity by showing a topological connection between the two notions of generalized continuity.

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Section: Introductionmentioning

confidence: 75%

“…But the inequality S = Y and the equality in (4) entail that z / ∈ B ca (x) for some z ∈ Y : an entailment that-by virtue of Lemma 2.1-is in contradiction with (1) and (5). Therefore, S = Y and hence {S} is a finite subcover of γ .…”

Section: Characterizations For Total Preordersmentioning

confidence: 97%

On the Existence of Greatest Elements and Maximizers

Quartieri

2022

J Optim Theory Appl

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A characterization of the generalized optimal choice set through the optimization of generalized weak utilities

Andrikopoulos

2015

Theory Decis

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A simple characterization of the existence of upper semicontinuous order-preserving functions

Bosi

Franzoi

2023

Econ Theory Bull

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We introduce an upper semicontinuity condition concerning a not necessarily total preorder on a topological space, namely strong upper semicontinuity, and in this way we extend to the nontotal case the famous Rader’s theorem, which guarantees the existence of an upper semicontinuous order-preserving function for an upper semicontinuous total preorder on a second countable topological space. We show that Rader’s theorem is not generalizable if we adopt weaker upper semicontinuity conditions already introduced in the literature. We characterize the existence of an upper semicontinuous order-preserving function for all strongly upper semicontinuous preorders on a metrizable topological space.

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A topological characterization of the existence of non-empty choice sets

Cited by 4 publications

References 19 publications

On the Existence of Greatest Elements and Maximizers

On the Existence of Greatest Elements and Maximizers

A characterization of the generalized optimal choice set through the optimization of generalized weak utilities

A simple characterization of the existence of upper semicontinuous order-preserving functions

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