An Alternative Description of Approach Spaces Via Approach Cores

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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“…For more details on the core we refer to [22]. The category App constitutes a framework wherein other important categories can be fully embedded.…”

Section: Preliminaries On Approach Spacesmentioning

confidence: 99%

“…Sometimes the structures on domain or codomain will not be mentioned explicitely. Then we have [22] (2.10)…”

Section: Preliminaries On Approach Spacesmentioning

confidence: 99%

Normality in terms of distances and contractions

Colebunders

Sioen

Haute

2018

Journal of Mathematical Analysis and Applications

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“…Approach spaces were introduced in 1989 by R. Lowen as a unification of metric spaces and topological spaces ( [L89], [L97], [L15], [SV16]). Instead of quantifying the properties of a space X by means of one metric, an approach space is determined by assigning to each point x ∈ X a collection A x of [0, ∞]-valued maps on X which are interpreted as local distances based at x.…”

Section: Introductionmentioning

confidence: 99%

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

Berckmoes¹,

Hellemans²,

Sioen³

et al. 2017

Journal of Mathematical Analysis and Applications

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Abstract. After shortly reviewing the fundamentals of approach theory as introduced by R. Lowen in 1989, we show that this theory is intimately related with the well-known Wasserstein metric on the space of probability measures with a finite first moment on a complete and separable metric space. More precisely, we introduce a canonical approach structure, called the contractive approach structure, and prove that it is metrized by the Wasserstein metric. The key ingredients of the proof of this result are Dini's Theorem, Ascoli's Theorem, and the fact that the class of real-valued contractions on a metric space has some nice stability properties. We then combine the obtained result with Prokhorov's Theorem to establish inequalities between the relative Hausdorff measure of non-compactness for the Wasserstein metric and a canonical measure of non-uniform integrability.

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“…(3) ⇒ (1): If (X, q) is a metric space, the set of all bounded lower regular functions L X coincides with the the set of all bounded contractions K b (X), [27]. Moreover (X, δ q ) is uniform and T 1 .…”

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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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An Alternative Description of Approach Spaces Via Approach Cores

Cited by 7 publications

References 10 publications

Normality in terms of distances and contractions

Normality in terms of distances and contractions

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

An isomorphism of the Wallman and Čech-Stone compactifications

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