Properties of several metric spaces of fuzzy sets

Huan Huang

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arXiv · 2304.03876

Properties of several metric spaces of fuzzy sets

Huan Huang

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Abstract

This paper discusses the properties the spaces of fuzzy sets in a metric space equipped with the endograph metric and the sendograph metric, respectively. We first give some relations among the endograph metric, the sendograph metric and the $\Gamma$-convergence, and then investigate the level characterizations of the endograph metric and the $\Gamma$-convergence. By using the above results, we give some relations among the endograph metric, the sendograph metric, the supremum metric and the $d_p^*$ metric, $p\geq 1$. On the basis of the above results, we present the characterizations of total boundedness, relative compactness and compactness in the space of fuzzy sets whose $\alpha$-cuts are compact when $\alpha>0$ equipped with the endograph metric, and in the space of compact support fuzzy sets equipped with the sendograph metric, respectively. Furthermore, we give completions of these metric spaces, respectively.

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Huan Huang. 2023-04-08. Properties of several metric spaces of fuzzy sets. https://arxiv.org/abs/2304.03876

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