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A natural topology for fuzzy numbers. (English) Zbl 1020.54005

A generalized fuzzy number is an upper semicontinuous function \(\mu: \overline{\mathbb{R}}\to [0,1]\) which attains the value 1 and is such that \(\mu(t)\geq \max\{\mu(s), \mu(r)\}\) whenever \(s\leq t\leq r\). A bijective correspondence between the set of generalized fuzzy numbers and the product \(\Delta\times\Delta\) (\(\Delta\) is the family of all distribution functions on \(\overline{\mathbb{R}}\)) is established. The set of generalized fuzzy numbers is then topologized by a kind of weak convergence (based on the weak convergence on \(\Delta\)) which turns it into a compact metric space whose topology is shown to be the interval topology induced by the pointwise ordering. A relationship between these fuzzy numbers and the \([0,1]\)-valued version of Hutton’s unit interval is discussed.

MSC:

54A40 Fuzzy topology
03E72 Theory of fuzzy sets, etc.
26E50 Fuzzy real analysis
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References:

[1] Gierz, G., A Compendium of Continuous Lattices (1980), Springer-Verlag: Springer-Verlag Berlin · Zbl 0452.06001
[2] Goetschel, R.; Voxman, W., Topological properties of fuzzy numbers, Fuzzy Sets and Systems, 10, 87-99 (1983) · Zbl 0521.54001
[3] Hutton, B., Normality in fuzzy topological spaces, J. Math. Anal. Appl., 50, 74-79 (1975) · Zbl 0297.54003
[4] Hutton, B., Uniformities on fuzzy topological spaces, J. Math. Anal. Appl., 58, 557-571 (1977) · Zbl 0358.54008
[5] Hutton, B.; Reilly, I. L., Separation axioms in fuzzy topological spaces, Fuzzy Sets and Systems, 3, 93-104 (1980) · Zbl 0421.54006
[6] Kaleva, O.; Seikkala, S., On fuzzy metric spaces, Fuzzy Sets and Systems, 12, 215-229 (1984) · Zbl 0558.54003
[7] Kubiak, T.; Zhang, D., On the \(L\)-fuzzy Brouwer fixed point theorem, Fuzzy Sets and Systems, 105, 287-292 (1999) · Zbl 0932.54011
[8] Loéve, M., Probability Theory (1977), Springer-Verlag: Springer-Verlag New York · Zbl 0359.60001
[9] Schweizer, B.; Sklar, A., Probabilistic Metric Spaces (1983), North-Holland: North-Holland New York · Zbl 0546.60010
[10] Wang, Guojun; Xu, Luoshan, Intrinsic topology and refinement of the Hutton unit interval, Sci. China Ser. A, 35, 1434-1443 (1992) · Zbl 0783.54008
[11] Zhang, Dexue, Metrizable completely distributive lattices, Comment. Math. Univ. Carolin., 31, 137-148 (1997) · Zbl 0887.06006
[12] Zhang, Dexue; Liu, Yingming, \(L\)-fuzzy modification of completely distributive lattices, Math. Nachr., 168, 79-95 (1994) · Zbl 0836.06008
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