##
**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.

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.

Reviewer: S.Gottwald (Leipzig)

### 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 |