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Compactness and convexity on the space of fuzzy sets. II. (English) Zbl 1065.54001
Let $${\mathcal I}(R^p)$$ denote the family of all fuzzy sets $$\widetilde u: \mathbb R^p\to [0,1]$$ with the following properties:
1) $$\widetilde u$$ is normal, i.e., there exists $$x\in \mathbb R^p$$ such that $$\widetilde u(x)= 1$$,
2) $$\widetilde u$$ is upper semicontinuous;
3) $$\text{supp\,}\widetilde u= \overline{\{x\in \mathbb R^p:\widetilde u(x)> 0\}}$$ is compact.
For introducing a topology in $${\mathcal I}(\mathbb R^p)$$ the following are required:
(i) $$P(\mathbb R^p)$$ denotes the family of non-empty compact substs of the Euclidean space $$\mathbb R^p$$ and the space is metrized by the Hausdorff metric $d_H(A,B)= \max \Bigl\{\sup_{a\in A} \,\inf_{b\in B}|a- b|,\;\sup_{b\in B}\, \inf_{a\in A}|a- |\Bigr\},$ where $$|.|$$ denotes the Euclidean norm;
(ii) For $$u\in \mathbb R^p$$, $$L_\alpha(\widetilde u)= \begin{cases} x:\widetilde u(x)\geq \alpha\;&\text{if }0< \alpha\leq 1,\\ \text{supp\,}\widetilde u\;&\text{if }\alpha= 0,\end{cases}$$;
(iii) $$d_\infty(\widetilde u,\widetilde v)= \sup_{0\leq \alpha\leq 1}\,d_H(L_\alpha\widetilde u, L_\alpha\widetilde v)$$.
The topology on $${\mathcal I}(\mathbb R^p)$$ is given by a metric $$d_s$$ which is defined as follows: with $$T$$ denoting the class of strictly increasing, continuous mappings of $$[0,1]$$ onto itself, for $$\widetilde u,\widetilde v\in{\mathcal I}(\mathbb R^p)$$, set $$d_s(\widetilde u,\widetilde v)= \text{inf}\{\varepsilon> 0$$: there exists $$t\in T$$ such that $$\sup_{0\leq\alpha\leq 1}|t_a\alpha- \alpha|\leq \varepsilon$$ and $$d_\infty(\widetilde u,t(\widetilde v))\leq \varepsilon\}$$, where $$t(\widetilde v)$$ denotes the composition of $$\widetilde v$$ and $$t$$. The author establishes some characterizations of convex and relatively compact subsets of the space $${\mathcal I}(\mathbb R^p)$$; the relationship between convergence in the $$d_s$$-metric and the $$d_\infty$$-metric is also investigated.
For part I of the paper see [J. Math. Anal. Appl. 264, 122–132 (2001; Zbl 1002.54005)].

##### MSC:
 54A40 Fuzzy topology 47H04 Set-valued operators 52A22 Random convex sets and integral geometry (aspects of convex geometry) 53C65 Integral geometry 54C60 Set-valued maps in general topology 46S40 Fuzzy functional analysis
Zbl 1002.54005
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