We introduce and discuss recent type of fibrewise topological spaces, namely fibrewise soft bitopological spaces. Also, we introduce the concepts of fibrewise closed soft bitopological spaces, fibrewise open soft bitopological spaces, fibrewise locally sliceable soft bitopological spaces and fibrewise locally sectionable soft bitopological spaces. Furthermore, we state and prove several propositions concerning these concepts.

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.

Abstract. One of the fibrewise micro-topological space is one in which the topology is decided through a group of fibre bundles, in comparison to the usual case in normal, fibrewise topological space. The micro-topological spaces draw power from their ability to be used in descriptions of a wide range of mathematical objects. These can be used to describe the topology of a manifold or even the topology of a group. Apart from easy manipulation, the fibrewise micro-topological spaces yield various mathematical applications, but the one being mentioned here is the possibility for geometric investigation of space or group structure. In this essay, we shall explain what fibrewise micro-topological spaces are, indicate why they are useful in math

The aim of this paper is to introduce and study some of the Fibrewise minimal regular,Fibrewise maximal regular, Fibrewise minimal completely regular, Fibrewise maximal completely regular, Fibrewise minimal normal, Fibrewise maximal normal, Fibrewise minimal functionally normal, and Fibrewise maximal functionally normal. This is done by providing some definitions of the concepts and examples related to them, as well as discussing some properties and mentioning some explanatory diagrams for those concepts.

Let R be a commutative ring with identity 1 ¹ 0, and let M be a unitary left module over R. A submodule N of an R-module M is called essential, if whenever N ⋂ L = (0), then L = (0) for every submodule L of M. In this case, we write N ≤e M. An R-module M is called extending, if every submodule of M is an essential in a direct summand of M. A submodule N of an R-module M is called semi-essential (denoted by N ≤sem M), if N ∩ P ≠ (0) for each nonzero prime submodule P of M. The main purpose of this work is to determine and study two new concepts (up to our knowledge) which are St-closed submodules and semi-extending modules. St-closed submodules is contained properly in the class of closed submodules, where a submodule N of

     In this paper, we study a new concept of fuzzy sub-module, called  fuzzy socle semi-prime sub-module that is a generalization the concept of semi-prime fuzzy sub-module and fuzzy of approximately semi-prime sub-module in the ordinary sense.  This leads us to introduce level property which studies the relation between the ordinary and fuzzy sense of approximately semi-prime sub-module. Also, some of its characteristics and notions such as the intersection, image and external direct sum of fuzzy socle semi-prime sub-modules are introduced. Furthermore, the relation between the fuzzy socle semi-prime sub-module and other types of fuzzy sub-module presented.

In this work the concept of semi-generalized regular topological space was introduced and studied via semi generalized open sets. Many properties and results was investigated and studied, also it was shown that the quotient space of semi-generalized regular topological space is not, in general semi-generalizedspace.

Dans la langue française, une forme d'auxiliarité, composée de deux éléments cohérents l'auxiliant et l'auxilié, fournit, en effet, à la phrase une diversité significative et structurale. L'auxiliarité, renvoie à l'unification de deux éléments grammaticaux afin de localiser l'énoncé sur l'axe du temps, d'aspect ou de mode. É. Benveniste définit l'auxiliarité en : « Il s'agit d'une forme linguistique unitaire qui se réalise, à travers des paradigmes entiers, en  deux éléments, dont chacun assume une partie des fonctions grammaticales, et qui sont à la fois liés et autonomes, distincts et complémentaires »[1]. Ces deux éléments d'auxiliarité possèden