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 primary objective of this paper is to introduce a new concept of fibrewise topological spaces on D is named fibrewise multi- topological spaces on D. Also, we entroduce the concepts of multi-proper, fibrewise multi-compact, fibrewise locally multi-compact spaces, Moreover, we study relationships between fibrewise multi-compact (resp., locally multi-compac) space and some fibrewise multi-separation axioms.

In this paper we define and study new concepts of fibrwise totally topological spaces over B namely fibrewise totally compact and fibrwise locally totally compact spaces, which are generalization of well known concepts totally compact and locally totally compact topological spaces. Moreover, we study relationships between fibrewise totally compact (resp, fibrwise locally totally compact) spaces and some fibrewise totally separation axioms.

  In this paper, we give the concept of N-open set in bitopological spaces, where N is the first letter of the name of one of the authors, then we used this concept to define a new kind of compactness, namely N-compactness and we define the N-continuous function in bitopological spaces.         We study some properties of N-compact spaces, and the relationships between this kind and two other known kinds which are S-compactness and pair-wise compactness.

In this paper we define and study new concepts of fibrewise topological spaces over B namely, fibrewise Lindelöf and locally Lindelöf topological spaces, which are generalizations of will-known concepts: Lindelöf topological space (1) "A topological space X is called a Lindelöf space if for every open cover of X has a countable subcover" and locally Lindelöf topological space (1) "A topological space X is called a locally Lindelöf space if for every point x in X, there exist a nbd U of x such that the closure of U in X is Lindelöf space". Either the new concepts are: "A fibrewise topological space X over B is called a fibrewise Lindelöf if the projection function p : X→B is Lindelöf" and "The fibrewise topological space X over B

This paper is devoted to introduce weak and strong forms of fibrewise fuzzy ω-topological spaces, namely the fibrewise fuzzy -ω-topological spaces, weakly fibrewise fuzzy -ω-topological spaces and strongly fibrewise fuzzy -ω- topological spaces. Also, Several characterizations and properties of this class are also given as well. Finally, we focused on studying the relationship between weakly fibrewise fuzzy -ω-topological spaces and strongly fibrewise fuzzy -ω-topological spaces.

In this paper we define and study new concepts of functions on fibrewise  topological spaces  over  B namely,  fibrewise  weakly  (resp., closure, strongly) continuoac; funttions which are analogous of weakly

(resp., closure,   strongly)   continuous  functions   and  the  main  result is   : Let  <p       : XY  be a fibrewise  closure  (resp.,  weakly,  closure, strongly,  strongly) continuous function,  where  Y  is fibrewise topological  space  over  B and  X  is  a  fibrewise set  which  has  the

in

In this thesis, we introduced some kinds of fibrewise topological spaces by using totally continuous function is called fibrewise totally topological spaces. We generalize some fundamental results from fibrewise topology into fibrewise totally topological spaces. We also introduce the concepts of fibrewise totally separation axioms, fibrewise totally compact and locally totally compact topological spaces. As well as fibrewise totally perfect topological spaces. We explain and discuss new notion of fibrewise topological spaces, namely fibrewise totally topological spaces. We, also introduce the concepts of fibrewise totally closed topological spaces, fibrewise totally open topological spaces, fibrewise locally sliceable and locally s

The aim of this paper is to introduce and study the notion type 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}.