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Characterizations of compact sets in fuzzy set spaces with L-p metric

机译：具有L-p度量的模糊集空间中紧集的特征

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Compactness criteria in fuzzy set spaces endowed with the L-p metric have been studied for several decades. Total boundedness is a key feature of compactness in metric spaces. However, comparing existing compactness criteria in fuzzy set spaces endowed with the L-p metric with the Arzela-Ascoli theorem, the latter gives compactness criteria by characterizing totally bounded sets while the former does not characterize totally bounded sets. Currently, compactness criteria are only presented for three particular fuzzy set spaces under assumptions of convexity or star-shapedness. General fuzzy sets have become more important in both theory and applications. Therefore, this paper presents characterizations of totally bounded sets, relatively compact sets, and compact sets in general fuzzy set spaces equipped with the L-p metric, but which do not have any assumptions of convexity or star-shapedness. Subsets of these general sets include common fuzzy sets, such as fuzzy numbers, fuzzy star-shaped numbers with respect to the origin, fuzzy star-shaped numbers, and general fuzzy star-shaped numbers. Existing compactness criteria are stated for fuzzy numbers space, the space of fuzzy star-shaped numbers with respect to the origin, and the space of fuzzy star-shaped numbers endowed with the L-p metric, respectively. Constructing completions of fuzzy set spaces with respect to the L-p metric is a problem closely dependent on characterizing totally bounded sets. Based on characterizations of total boundedness and relatively compactness and some discussion of the convexity and star-shapedness of fuzzy sets, we show that the completions of fuzzy set spaces studied here can be obtained using the L-p extension. We also clarify relationships among the ten fuzzy set spaces studied here-the five pairs of original spaces and their corresponding completions. We show that the subspaces have parallel characterizations of totally bounded sets, relatively compact sets, and compact sets. Finally, we discuss properties of the L-p metric on fuzzy set space as an application of our results, and review compactness criteria proposed in previous work. (C) 2016 Elsevier B.V. All rights reserved.
机译：具有L-p度量的模糊集空间中的紧致性准则已经研究了数十年。总有界性是度量空间中紧凑性的关键特征。但是，将具有L-p度量的模糊集空间中的现有紧致度标准与Arzela-Ascoli定理进行比较，后者通过表征完全有界集来给出紧致度标准，而前者没有表征完全有界集。当前，仅在凸度或星形的假设下针对三个特定的模糊集空间提出了压缩标准。通用模糊集在理论和应用中都变得越来越重要。因此，本文介绍了具有L-p度量的一般模糊集空间中的完全有界集，相对紧集和紧集的特征，但没有任何凸度或星形的假设。这些通用集的子集包括通用模糊集，例如模糊数，相对于原点的模糊星形数，模糊星形数和通用模糊星形数。针对模糊数空间，相对于原点的模糊星形数空间以及赋予L-p度量的模糊星形数空间规定了现有的紧致性准则。关于L-p度量构造模糊集空间的完成是一个紧密依赖于表征完全有界集的问题。基于总有界性和相对紧性的特征，以及对模糊集的凸性和星形的讨论，我们表明可以使用L-p扩展获得本文研究的模糊集空间的完备性。我们还将弄清这里研究的十个模糊集空间之间的关系，即五对原始空间及其对应的完成度。我们证明子空间具有完全有界集，相对紧集和紧集的并行表征。最后，我们讨论模糊集空间上的L-p度量的属性作为我们的结果的应用，并回顾先前工作中提出的紧缩性准则。 （C）2016 Elsevier B.V.保留所有权利。

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