The objective of this paper is to define and introduce a new type of nano semi-open set which called nano -open set as a strong form of nano semi-open set which is related to nano closed sets in nano topological spaces. In this paper, we find all forms of the family of nano -open sets in term of upper and lower approximations of sets and we can easily find nano -open sets and they are a gate to more study.  Several types of nano open sets are known, so we study relationship between the nano -open sets with the other known types of nano open sets in nano topological spaces. The Operators such as nano -interior and nano -closure are the part of this paper.

We define and study new ideas of fibrewise topological space on D namely fibrewise multi-topological space on D. We also submit the relevance of fibrewise closed and open topological space on D. Also fibrewise multi-locally sliceable and fibrewise multi-locally section able multi-topological space on D. Furthermore, we propose and prove a number of statements about these ideas.

The essential objective of this paper is to introduce new notions of fibrewise topological spaces on D that are named to be upper perfect topological spaces, lower perfect topological spaces, multi-perfect topological spaces, fibrewise upper perfect topological spaces, and fibrewise lower perfect topological spaces. fibrewise multi-perfect topological spaces, filter base, contact point, rigid, multi-rigid, multi-rigid, fibrewise upper weakly closed, fibrewise lower weakly closed, fibrewise multi-weakly closed, set, almost upper perfect, almost lower perfect, almost multi-perfect, fibrewise almost upper perfect, fibrewise almost lower perfect, fibrewise almost multi-perfect, upper^* continuous fibrewise upper^∗ topol

        Throughout this paper, we introduce the notion of weak essential F-submodules of F-modules as a generalization of  weak essential submodules. Also we study the homomorphic image and inverse image of weak essential F-submodules.

The primary objective of this paper is to present a new concept of fibrewise topological spaces over B is said to be fibrewise slightly topological spaces over B. Also, we introduce the concepts of fibrewise slightly perfect topological spaces, filter base, contact point, slightly convergent, slightly directed toward a set, slightly adherent point, slightly rigid, fibrewise slightly weakly closed, H.set, fibrewise almost slightly perfect, slightly∗ .continuous fibrewise slightly∗ topological spaces respectively, slightly Te, locally QHC, In addition, we state and prove several propositions related to these concepts.

            In this paper the definition of fuzzy anti-normed linear spaces and its basic properties are used to prove some properties of a finite dimensional fuzzy anti-normed linear space.    

The concept of Cech fuzzy soft bi-closure space ( ˇ Cfs bi-csp) ( ˇ U, L1, L2, S) is initiated and studied by the authors in [6]. The notion of pairwise fuzzy soft separated sets in Cfs bi-csp is defined in this study, and various features of ˇ this notion are proved. Then, we introduce and investigate the concept of connectedness in both Cfs bi-csps and its ˇ associated fuzzy soft bitopological spaces utilizing the concept of pairwise fuzzy soft separated sets. Furthermore, the concept of pairwise feebly connected is introduced, and the relationship between pairwise connected and pairwise feebly connected is discussed. Finally, we provide various instances to further explain our findings.

  In this paper we introduce the notion of semiprime fuzzy module as a generalization of semiprime module. We investigate several characterizations and properties of this concept.

         In this paper, developed Jungck contractive mappings into fuzzy Jungck contractive and proved  fuzzy fixed point for some types of generalize fuzzy Jungck contractive mappings.