This paper introduces some properties of separation axioms called α -feeble regular and α -feeble normal spaces (which are weaker than the usual axioms) by using elements of graph which are the essential parts of our α -topological spaces that we study them. Also, it presents some dependent concepts and studies their properties and some relationships between them.

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 aim of this research is to study some types of fibrewise fuzzy topological spaces. The six major goals are explored in this thesis. The very first goal, introduce and study the notions types of fibrewise topological spaces, namely fibrewise fuzzy j-topological spaces, Also, we introduce the concepts of fibrewise j-closed fuzzy topological spaces, fibrewise j-open fuzzy topological spaces, fibrewise locally sliceable fuzzy j-topological spaces and fibrewise locally sectionable fuzzy j-topological spaces. Furthermore, we state and prove several Theorems concerning these concepts, where j={δ,θ,α,p,s,b,β} The second goal is to introduce weak and strong forms of fibrewise fuzzy ω-topological spaces, namely the fibrewise fuz

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

     In this paper we introduce a new class of sets called -generalized b- closed (briefly gb closed) sets. We study some of its basic properties. This class of sets is strictly placed between the class of gp- closed sets and the class of gsp- closed sets. Further the notion of b- space is introduced and studied.

2000 Mathematics Subject Classification: 54A05

We define and study new ideas of fibrewise topological space namely fibrewise multi-topological space . We also submit the relevance of fibrewise closed and open topological space . Also fibrewise multi-locally sliceable and fibrewise multi-locally section able multi-topological space . Furthermore, we propose and prove a number of statements about these ideas. On the other hand, extend separation axioms of ordinary topology into fibrewise setting. The separation axioms are said to be fibrewise multi-T0. spaces, fibrewise multi-T1spaces, fibrewise multi-R0 spaces, fibrewise multi-Hausdorff spaces, fibrewise multi-functionally Hausdorff spaces, fibrewise multi-regular spaces, fibrewise multi-completely regular spaces, fibrewise multi-normal