Fibrewise topological spaces theory is a relatively new branch of mathematics, less than three decades old, arisen from algebraic topology. It is a highly useful tool and played a pivotal role in homotopy theory. Fibrewise topological spaces theory has a broad range of applications in many sorts of mathematical study such as Lie groups, differential geometry and dynamical systems theory. Moreover, one of the main objects, which is considered in fibrewise topological spaces theory is connectedness. In this regard, we of the present study introduce the concept of connected fibrewise topological spaces and study their main results.

The purpose of this paper is to introduce and study the concepts of fuzzy generalized open sets, fuzzy generalized closed sets, generalized continuous fuzzy proper functions and prove results about these concepts.

The main idea of this research is to consider fibrewise pairwise versions of the more important separation axioms of ordinary bitopology named fibrewise pairwise - spaces, fibrewise pairwise - spaces, fibrewise pairwise - spaces, fibrewise pairwise -Hausdorff spaces, fibrewise pairwise functionally -Hausdorff spaces, fibrewise pairwise -regular spaces, fibrewise pairwise completely -regular spaces, fibrewise pairwise -normal spaces and fibrewise pairwise functionally -normal spaces. In addition we offer some results concerning it.

In this paper, a new type of supra closed sets is introduced which we called supra β*-closed sets in a supra topological space. A new set of separation axioms is defined, and its many properties are examined. The relationships between supra β*-Ti –spaces (i = 0, 1, 2)   are studied and shown with instances. Additionally, new varieties of supra β*-continuous maps have been taken into consideration based on the supra β*-open sets theory. 

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.

Form the series of generalization of the topic of supra topology is the generalization of separation axioms . In this paper we have been introduced (S * - SS *) regular spaces . Most of the properties of both spaces have been investigated and reinforced with examples . In the last part we presented the notations of supra *- -space ( =0,1) and we studied their relationship with (S * - SS *) regular spaces.

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