An application of approach theory to the relative Hausdorff measure of non-compactness for the Wasserstein metric

Berckmoes, Ben; Hellemans, Tim; Sioen, Mark; Casteren, J. van

doi:10.1016/j.jmaa.2017.01.008

Cited by 3 publications

(2 citation statements)

References 32 publications

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“…Such isometric settings get more and more attention like for instance in the study of approximation by Lipschitz functions in [13], of cofinal completeness and the UC-property in [2], in investigations on hyperconvexity in [19] and on the non-symmetric analogue of the Urysohn metric space in [20] and [21]. For other applications the larger context of approach spaces with contractions is even more suitable as was recently shown in the context of probability measures [3], [4] and [5], or complexity analysis [7] and [8].…”

Section: Introductionmentioning

confidence: 99%

An isomorphism of the Wallman and Čech-Stone compactifications

Colebunders

Löwen

Sioen

2021

Quaestiones Mathematicae

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For a metrizable topological space X it is well known that in general the Čech-Stone compactification β(X) or the Wallman compactification W (X) are not metrizable. To remedy this fact one can alternatively associate a point-set distance to the metric, a so called approach distance. It is known that in this setting both a Čech-Stone compactification β * (X) and a Wallman compactification W * (X) can be constructed in such a way that their approach distances induce the original approach distance of the metric on X [23], [24].The main goal in this paper is to formulate necessary and sufficient conditions for an approach space X such that the Čech-Stone compactification β * (X) and the Wallman compactification W * (X) are isomorphic, thus answering a question first raised in [24]. The first clue to reach this goal is to settle a question left open in [10], to formulate sufficient conditions for a compact approach space to be normal. In particular the result shows that the Čech-Stone compactification β * (X) of a uniform T 2 space, is always normal. We prove that the Wallman compactification W * (X) is normal if and only if X is normal, and we produce an example showing that, unlike for topological spaces, in the approach setting normality of X is not sufficient for β * (X) and W * (X) to be isomorphic. We introduce a strengthening of the regularity condition on X, which we call ideal-regularity, and in our main theorem we conclude that X is ideal-regular, normal and T 1 if and only if X is a uniform T 1 approach space with β * (X) and W * (X) isomorphic. Classical topological results are recovered and implications for (quasi-)metric spaces are investigated.

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Section: Introductionmentioning

confidence: 99%

An isomorphism of the Wallman and Čech-Stone compactifications

Colebunders

Löwen

Sioen

2021

Quaestiones Mathematicae

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“…Such isometric settings get more and more attention like for instance in the study of approximation by Lipschitz functions in [12], of cofinal completeness and UC-property in [1], in investigations on hyperconvexity in [15] and on the non-symmetric analogue of the Urysohn metric space in [16] and [17]. For other applications the larger context of approach spaces with contractions is even more suitable as was recently shown in the context of probability m easures [2], [3] and [4], or complexity analysis [6] and [7].…”

Section: Introductionmentioning

confidence: 99%

Normality in terms of distances and contractions

Colebunders

Sioen

Haute

2018

Journal of Mathematical Analysis and Applications

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The main purpose of this paper is to explore normality in terms of distances between points and sets. We prove some important consequences on realvalued contractions, i.e. functions not enlarging the distance, showing that as in the classical context of closures and continuous maps, normality in terms of distances based on an appropriate numerical notion of γ-separation of sets, has far reaching consequences on real valued contractive maps, where the real line is endowed with the Euclidean metric. We show that normality is equivalent to (1) separation of γ-separated sets by some Urysohn contractive map, (2) to Katětov-Tong's interpolation, stating that for bounded positive realvalued functions, between an upper and a larger lower regular function, there exists a contractive interpolating map and (3) to Tietze's extension theorem stating that certain contractions defined on a subspace can be contractively extended to the whole space.The appropriate setting for these investigations is the category of approach spaces, but the results have (quasi)-metric counterparts in terms of non-expansive maps. Moreover when restricted to topological spaces, classical normality and its equivalence to separation by a Urysohn continuous map, to Katětov-Tong's interpolation for semicontinuous maps and to Tietze's extension theorem for continuous maps are recovered.

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Supported approach spaces

Colebunders,

Lowen

2023

Quaestiones Mathematicae

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An application of approach theory to the relative Hausdorff measure of non-compactness for the Wasserstein metric

Cited by 3 publications

References 32 publications

An isomorphism of the Wallman and Čech-Stone compactifications

An isomorphism of the Wallman and Čech-Stone compactifications

Normality in terms of distances and contractions

Supported approach spaces

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