The purpose of this paper is to consider fibrewise near versions of the more important separation axioms of ordinary topology namely fibrewise near T0 spaces, fibrewise near T1 spaces, fibrewise near R0 spaces, fibrewise near Hausdorff spaces, fibrewise near functionally Hausdorff spaces, fibrewise near regular spaces, fibrewise near completely regular spaces, fibrewise near normal spaces and fibrewise near functionally normal spaces. Also we give several results concerning it.

   The aim of this paper is to introduce and study new class of fuzzy function called fuzzy semi pre homeomorphism in a fuzzy topological space by utilizing fuzzy semi pre-open sets. Therefore, some of their characterization has been proved; In addition to that we define, study and develop corresponding to new class of fuzzy semi pre homeomorphism in fuzzy topological spaces using this new class of functions.

In this paper we introduce a lot of concepts in bitopological spaces which are ij-ω-converges to a subset, ij-ω-directed toward a set, ij-w-closed functions, ij-w-rigid set, ij-w-continuous functions and the main concept in this paper is ij-w-perfect functions between bitopological spaces. Several theorems and characterizations concerning these concepts are studied.

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.

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.