## Metric spaces of fuzzy sets: theory and applications.(English)Zbl 0873.54019

Singapore: World Scientific. ix, 178 p. (1994).
This is a mathematical study of spaces of fuzzy sets, mainly fuzzy subsets of $$\mathbb{R}^n$$. The introductory Chapter 1 is followed by four chapters which outline background material about metric spaces of nonempty compact or compact convex subsets of $$\mathbb{R}^n$$ and Banach spaces, and about set valued mappings of a real variable.
The next five chapters apply these ideas to metric spaces of fuzzy sets, essentially upper semicontinuous and often even convex fuzzy subsets of $$\mathbb{R}^n$$, which hence have compact or compact convex $$\alpha$$-level sets. These metric spaces are studied in detail together with fuzzy set valued mappings of real variables.
The remaining five chapters are devoted to applications of the previously proven results: to fuzzy random variables; to computational methods like least squares, fuzzy kriging and interpolation; to fuzzy differential equations; to optimization under uncertainty; and to fuzzy iterations and image processing.
An appendix which presents the basic facts on metric spaces, an index, and a list of symbols close this well written interesting book.

### MSC:

 54C35 Function spaces in general topology 54E35 Metric spaces, metrizability 54B20 Hyperspaces in general topology 03E72 Theory of fuzzy sets, etc. 26E50 Fuzzy real analysis 46S40 Fuzzy functional analysis 26E25 Set-valued functions

Zbl 0873.54007