Here, we deal with the concept of fuzzy metric space $(\mathcal{X},\mathcal{M},*)$ , due to George and Veeramani. Based on the fuzzy diameter for a subset of $\mathcal{X}$ , we introduce the notion of strong fuzzy diameter zero for a family of subsets. Then, we characterize nested sequences of subsets having strong fuzzy diameter zero using their fuzzy diameter. Examples of sequences of subsets which do or do not have strong fuzzy diameter zero are provided. Our main result is the following characterization: a fuzzy metric space is strongly complete if and only if every nested sequence of close subsets which has strong fuzzy diameter zero has a singleton intersection. Moreover, the standard fuzzy metric is studied as a particular case. Finally, this work points out a route of research in fuzzy fixed point theory.

中文翻译：

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模糊度量空间中强完备性的刻画

在这里，我们处理模糊度量空间的概念 $（\mathcal{X}，\mathcal{中号}，*）$ 由于George和Veeramani。基于模糊直径的子集 $\mathcal{X}$ ，我们为一族子集引入了强模糊直径零的概念。然后，我们使用模糊直径来描述具有强模糊直径为零的子集的嵌套序列。提供了不具有强模糊直径零的子集序列的示例。我们的主要结果是以下表征：当且仅当具有强模糊直径零的封闭子集的每个嵌套序列具有单例交集时，模糊度量空间才是完全完备的。此外，作为特殊情况，研究了标准模糊度量。最后，本文指出了模糊定点理论的研究路径。