On I-topology generated by fuzzy norm

A new I-vector topology generated by a fuzzy norm

“…Definition 2.6 (Fang [6]). A fuzzy set on (X, · , L, R) is said to be · -strongly open if for every x ∈ supp and ∈ (0, (x)), there exists ε > 0 such that (x, ε) ⊂ .…”

“…Recently, Fang [6] pointed out that the Das's I-topological structure on the fuzzy normed linear space is incompatible with the linear structure, that is, the fuzzy normed linear space is not an I-topological vector space with respect to the Das's I-topology. In order to overcome this incompatibility, Fang [6] constructed another I-topology on the fuzzy normed linear space by changing the definition of open fuzzy subset given by Das [2], and proved that the fuzzy normed linear space is a Hausdorff locally convex I-topological vector space with respect to this I-topology under certain conditions. In this paper, by using an approach different from [6], we construct a new I-vector topology T · on the fuzzy normed linear space and study some of its properties.…”

“…In order to overcome this incompatibility, Fang [6] constructed another I-topology on the fuzzy normed linear space by changing the definition of open fuzzy subset given by Das [2], and proved that the fuzzy normed linear space is a Hausdorff locally convex I-topological vector space with respect to this I-topology under certain conditions. In this paper, by using an approach different from [6], we construct a new I-vector topology T · on the fuzzy normed linear space and study some of its properties. In Section 3, we prove that under proper condition, the fuzzy normed linear space is not only a locally convex I-topological vector space but of QL-type with respect to the I-topology T · .…”

“…We also give a necessary and sufficient condition for the I-vector topology T · to be Hausdorff and weakly Hausdorff, respectively. In Section 4, we discuss the relations between I-vector topologies T · and other two I-vector topologies T * · and ( ) on the fuzzy normed linear space, where T * · is defined in [6] and ( ) is the induced I-topology of the crisp vector topology determined by fuzzy norm · (see [5] or [16]). We prove a necessary and sufficient condition for the equality T · = T * · to hold.…”

Fundamental Stabilities of the Nonic Functional Equation in Intuitionistic Fuzzy Normed Spaces

“…Then, Saadati et al have obtained a modified case of intuitionistic fuzzy normed spaces by improving the separation condition and strengthening some conditions in the definition of [24]. Many authors have considered the intuitionistic fuzzy normed linear spaces, and intuitionistic fuzzy 2-normed spaces (see [1], [2], [13], [16]). Also, the generalized Hyers-Ulam stability of different functional equations in intuitionistic fuzzy normed spaces has been studied by a number of the authors (see [5], [15], [20], [21] and [26])…”