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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
Citations:
Zbl 1002.54005
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References:
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