Abstract This paper is devoted to introduce weak and strong forms of fibrewise fuzzy u-topological spaces, namely the fibrewise fuzzy q-u-topological spaces, weakly fibrewise fuzzy q-u-topological spaces and strongly fibrewise fuzzy q-utopological 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 q-u-topological spaces and strongly fibrewise fuzzy q-utopological spaces.

This paper is concerned with introducing and studying the new approximation operators based on a finite family of d. g. 'swhich are the core concept in this paper. In addition, we study generalization of some Pawlak's concepts and we offer generalize the definition of accuracy measure of approximations by using a finite family of d. g. 's.

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.

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

In the present paper, a simply* compact spaces was introduced it defined over simply*- open set previous knowledge and we study the relation between the simply* separation axioms and the compactness, in addition to introduce a new types of functions known as 𝛼𝑆 𝑀∗ _irresolte , 𝛼𝑆 𝑀∗ __𝑐𝑜𝑛𝑡𝑖𝑛𝑢𝑜𝑢𝑠 and 𝑅 𝑆 𝑀∗ _ continuous, which are defined between two topological spaces.