Weakly induced modifications of I-fuzzy topologies

Since Chang [2] introduced fuzzy theory into topology, many authors have discussed various aspects of fuzzy topology. It is well known that weakly induced and induced topological spaces play an important role in L-topological spaces (see book [8]). According to their value ranges, L-topological spaces form different categories. Clearly, the investigation on their relationships is certainly important and necessary. Lowen was the first author to study the relations between I-topological spaces and classical topological spaces. He introduced two well-known functors—ω and ι. Later, these functors, named Lowen functors, were extended by different authors [7, 12] for various kinds of lattices studying the relations between L-TOP and TOP. However, in a completely different direction, Höhle [4] created the notion of a topology being viewed as an L-subset of a powerset. Then Kubiak [6] and Šostak [11] independently extended Höhle’s notion to L-subsets of LX . From a logical point of view, Ying [13] introduced fuzzifying topological spaces (Ying’s fuzzifying topology is similar to Höhle’s topology). In order to discuss the relations between fuzzifying topologies and I-fuzzy topologies, the authors studied Lowen functors in I-fuzzy topological spaces in a KubiakŠostak sense and introduced induced I-fuzzy topological spaces in [15]. Zhang and Liu [17] studied weakly induced modifications of L-topologies. The aim of this paper is to study weakly induced I-fuzzy topological spaces and the weakly induced modifications of I-fuzzy topologies. This paper is organized as follows. In Section 1, we give some preliminary concepts and properties. Two kinds of weakly induced modifications are introduced in Section 2.