A fuzzy compactum in a space X is an upper semi-continuous map $f:A\to [0,1]$ from a nonempty compact subspace $A\subset X$ to the unit closed interval $[0,1]$ such that $\{x\in A:f(x)>0\}$ is dense in A. In this paper, we investigate the topological structure of the space ${\mathcal{K}}_e(X)$ of fuzzy compacta of a separable metric space X with the Hausdorff metric. In particular, we prove that ${\mathcal{K}}_e(X)\approx {\ell }^2$ if X is an infinite, locally compact, locally connected, connected and separable metric space.

中文翻译：

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模糊致密空间的拓扑结构

甲模糊致密的空间中X是一个上半连续地图$F：\mathrm{一种}\to [0,1]$ 来自非空紧致子空间 $\mathrm{一种}\subset X$ 到单位闭区间 $[0,1]$ 以至于 $\{X\in \mathrm{一种}：F(X)>0\}$在A 中是稠密的。在本文中，我们研究了空间的拓扑结构${\mathcal{钾}}_{\mathrm{电子}}(X)$具有 Hausdorff 度量的可分离度量空间X的模糊紧致。特别地，我们证明${\mathcal{钾}}_{\mathrm{电子}}(X)\approx {\ell }^2$如果X是一个无限的、局部紧的、局部连通的、连通的和可分的度量空间。