The Quasimetrization Problem in the (Bi)topological Spaces

Andrikopoulos, Athanasios

doi:10.1155/2007/76904

Cited by 5 publications

(4 citation statements)

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“…The problem of quasi-pseudo-metrization of a bitopological space was considered in Kelly's work [31] and has been extensively studied over the years [37,38,39,40,41,42,43,44]. For bitopological spaces Kelly [31, Theor.…”

Section: Strict Quasi-pseudo-metrization From the I-space Conditionmentioning

confidence: 99%

“…Let us recall that a topological preordered space is a regularly preordered space if it is semiclosed preordered, (a) for every closed decreasing set The problem of quasi-pseudo-metrization of a bitopological space was considered in Kelly's work [31] and has been extensively studied over the years [37,38,39,40,41,42,43,44]. For bitopological spaces Kelly [31, Theor.…”

Section: Strict Quasi-pseudo-metrization From the I-space Conditionmentioning

confidence: 99%

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Convexity and quasi-uniformizability of closed preordered spaces

Minguzzi

2013

Topology and its Applications

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In many applications it is important to establish if a given topological preordered space has a topology and a preorder which can be recovered from the set of continuous isotone functions. Under antisymmetry this property, also known as quasi-uniformizability, allows one to compactify the topological space and to extend its order dynamics. In this work we study locally compact $\sigma$-compact spaces endowed with a closed preorder. They are known to be normally preordered, and it is proved here that if they are locally convex, then they are convex, in the sense that the upper and lower topologies generate the topology. As a consequence, under local convexity they are quasi-uniformizable. The problem of establishing local convexity under antisymmetry is studied. It is proved that local convexity holds provided the convex hull of any compact set is compact. Furthermore, it is proved that local convexity holds whenever the preorder is compactly generated, a case which includes most examples of interest, including preorders determined by cone structures over differentiable manifolds. The work ends with some results on the problem of quasi-pseudo-metrizability. As an application, it is shown that every stably causal spacetime is quasi-uniformizable and every globally hyperbolic spacetime is strictly quasi-pseudo-metrizable.Comment: Latex2e, 25 pages. v2: Simplified and reorganized, added introductory section on causalit

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Section: Strict Quasi-pseudo-metrization From the I-space Conditionmentioning

confidence: 99%

Section: Strict Quasi-pseudo-metrization From the I-space Conditionmentioning

confidence: 99%

Convexity and quasi-uniformizability of closed preordered spaces

Minguzzi

2013

Topology and its Applications

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“…The problem of quasi-pseudo-metrization of a bitopological space was considered already in Kelly's work [10] and has been extensively studied over the years [17,18,19,20,21,22,23,24]. As we shall see in a moment, the solution to this problem can be used to infer results on the quasi-pseudo-metrizability of a topological preordered space.…”

Section: Definition 21mentioning

confidence: 99%

Quasi-pseudo-metrization of topological preordered spaces

Minguzzi¹

2012

Topology and its Applications

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We establish that every second countable completely regularly preordered space (E,T,\leq) is quasi-pseudo-metrizable, in the sense that there is a quasi-pseudo-metric p on E for which the pseudo-metric p\veep^-1 induces T and the graph of \leq is exactly the set {(x,y): p(x,y)=0}. In the ordered case it is proved that these spaces can be characterized as being order homeomorphic to subspaces of the ordered Hilbert cube. The connection with quasi-pseudo-metrization results obtained in bitopology is clarified. In particular, strictly quasi-pseudometrizable ordered spaces are characterized as being order homeomorphic to order subspaces of the ordered Hilbert cube.Comment: Latex2e, 20 pages. v2: minor changes in the proof of theorem 2.

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“…Preference orders arising in this fashion we call quasi-metrizable or simply metrizable. A review of different sets of necessary and sufficient conditions for quasi-metrizability can be found in Andrikopoulos (2007). Therefore, in case a number of topologies generating a given preference order appear to be quasi-metrizable, then there will be a number of quasisemidistances generating one and the same preference order.…”

Section: Preference Relations and Topologymentioning

confidence: 99%

Stochastic Dominance Revisited

Rachev

Stoyanov

Fabozzi

2011

A Probability Metrics Approach to Financial Risk Measures

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The goals of this chapter are the following:• To explore the relationship between preference relations and quasi-semidistances. • To introduce a universal description of probability quasisemidistances in terms of a Hausdorff structure. • To provide examples with first-, second-, and higher-order stochastic dominance and to introduce primary, simple, and compound stochastic orders. • To explore new stochastic dominance rules based on a popular risk measure. • To provide a utility-type representation of probability quasisemidistances and to describe the degree of violation utilized in almost stochastic orders in terms of quasi-semidistances.

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The Quasimetrization Problem in the (Bi)topological Spaces

Abstract: It is our main purpose in this paper to approach the quasi-pseudometrization problem in (bi)topological spaces in a way which generalizes all the well-known results on the subject naturally, and which is close to a "Bing-Nagata-Smirnov style" characterization of quasi-pseudometrizability.

Cited by 5 publications

References 19 publications

Convexity and quasi-uniformizability of closed preordered spaces

Convexity and quasi-uniformizability of closed preordered spaces

Quasi-pseudo-metrization of topological preordered spaces

Stochastic Dominance Revisited

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