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In this paper, some relations among the axioms of KM fuzzy metric spaces and KM fuzzy normed spaces are discussed, and some characterizations concerning the additional axiom are given. Also, a quasi-metric family and a quasi-norm family corresponding, respectively, to the KM fuzzy metric spaces and the KM fuzzy normed spaces are introduced and studied. Using their properties, the different topological structures of KM fuzzy metric spaces and KM fuzzy normed spaces with variable t-norm are described, and some new properties on these spaces are obtained. As their special cases, the topological structures and properties concerning GV fuzzy metric spaces and GV fuzzy normed spaces are presented. [ABSTRACT FROM AUTHOR]

Recently, a modified definition of a fuzzy inner product space was introduced by Saheli and Gelousalar (2016) [1]. Here we provide some comments to this paper. It is pointed out that the additivity axiom (FI7) is a strong condition which leads their setting to be the null space in reality, so the orthogonality of the two vectors is not discussed in general. [ABSTRACT FROM AUTHOR]

Abstract: In this paper level-cut measures of noncompactness are introduced in standard fuzzy normed spaces and condensing operators are studied. Using the extensions of nonlinear compact operators, topological degree theory with respect to condensing operators is given in standard fuzzy normed spaces. As an application of degree theory, a fixed point theorem for condensing operators is presented. [Copyright &y& Elsevier]

In this paper, the Leray–Schauder topological degree theory is developed in a fuzzy normed space. Since the linear topology on this fuzzy normed space is not necessarily locally convex, and since each Menger probabilistic normed space can be considered as a special fuzzy normed space, the degree theory in this paper is different from the degree theory in locally convex linear topological space presented by Nagumo (Amer. J. Math. 73 (1951) 497–511), and it also is an extension of the degree theory in Menger probabilistic normed space studied by Zhang and Chen (Appl. Math. Mech. 10(6) (1989) 477–486). Applying this degree theory, some fixed point theorems for operators are given in fuzzy normed spaces, and some former corresponding results are extended and improved. [Copyright &y& Elsevier]