Completeness, metrizability and compactness in spaces of fuzzy-number-valued functions

Abstract

Fuzzy-number-valued functions, that is, functions defined on a topological space taking values in the space of fuzzy numbers, play a central role in the development of Fuzzy Analysis. In this paper we study completeness, metrizability and compactness of spaces of continuous fuzzy-number-valued functions.

Answers to some questions about Zadeh's extension on metric spaces

“…Recently, Jardón et al [5] proved that both f and F are contractive if the previous dynamical systems are contractive under the level-wise metric d ∞ , and they proposed the following two questions. For more recent results on Zadeh's extension and g-fuzzification, refer to [2,3,7,8,9,10] and some references therein. Question 3.…”

Endograph Metric and a Version of the Arzelà–Ascoli Theorem for Fuzzy Sets

“…Actually, given a uniform space Y, an Arzelà-Ascoli-type theorem characterizes compactness in the function space C(X, Y) by means of equicontinuity plus natural conditions. In fuzzy analysis, examples of this situation are the Arzelà-Ascoli-type theorems presented in [17]: the authors characterize compact subsets of C τ α (X, (E 1 , d ∞ )), the space of all continuous functions from a Tychonoff space X into the space of real (compact) fuzzy numbers (endowed with the supremum distance) where τ α is the topology of the uniform convergence on the members of a cover α. The characterization is obtained in terms of equicontinuity plus fuzzy conditions.…”

Espacios de funciones continuas con valores difusos

“…En la primera parte de esta memoria, hemos caracterizado la compleción del espacio C τα (X, (E 1 , d ∞ )) basándonos en los α f -espacios y en los subconjuntos acotados. A continuación, caracterizamos la metrizabilidad de C τα (X, (E 1 , d ∞ )) a partir de la hemi-α-compacidad de X. Para terminar esta parte, probamos varios teoremas tipo Ascoli para espacios de funciones C τα (X, (E 1 , d ∞ )) y C co (X, (E 1 , τ )) ( [26]).…”