In this paper, we will introduce and study the concept of nano perfect mappings by using the definition of nano continuous mapping and nano closed mapping, study the relationship between them, and discuss them with many related theories and results. The k-space and its relationship with nano-perfect mapping are also defined.

The aims of this thesis are to study the topological space; we introduce a new kind of perfect mappings, namely j-perfect mappings and j-ω-perfect mappings. Furthermore, we devoted to study the relationship between j-perfect mappings and j-ω-perfect mappings. Finally, certain theorems and characterization concerning these concepts are studied. On the other hand, we studied weakly/ strongly forms of ω-perfect mappings, namely -ω-perfect mappings, weakly -ω-perfect mappings and strongly-ω-perfect mappings; also, we investigate their fundamental properties. We devoted to study the relationship between weakly -ω-perfect mappings and strongly -ω-perfect mappings. As well as, some new generalizations of some definitions wh

In this work, we introduce a new kind of perfect mappings, namely j-perfect mappings and j-ω-perfect mappings. Furthermore we devoted to study the relationship between j-perfect mappings and j-ω-perfect mappings. Finally, certain theorems and characterization concerning these concepts are studied; j = , δ, α, pre, b, β

In this paper the concepts of weakly (resp., closure, strongly) Perfect Mappings are defined and the important relationships are studied: (a) Comparison between deferent forms of perfect mappings. (b) Relationship between compositions of deferent forms of perfect mappings. (c) Investigate relationships between deferent forms of perfect mappings and their graphs mappings.

The aim of the research is to apply fibrewise multi-emisssions of the paramount separation axioms of normally topology namely fibrewise multi-T0. spaces, fibrewise multi-T1 spaces, 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 spaces and fibrewise multi-functionally normal spaces. Also we give many score regarding it.

We introduce some new generalizations of some definitions which are, supra closure converge to a point, supra closure directed toward a set, almost supra converges to a set, almost supra cluster point, a set supra H-closed relative, supra closure continuous functions, supra weakly continuous functions, supra compact functions, supra rigid a set, almost supra closed functions and supra perfect functions. And we state and prove several results concerning it

Abstract. In this work, some new concepts were introduced and the relationship between them was studied. These concepts are filter directed-toward, nano-closure-directed-toward and nano-closure-converge to point, and some theories and results about these concepts were presented. A definition almost-nano-converges for set, almost-nano-cluster-point, and definition of quasi-nano-Hausdorff-closed and was also called nano-Hausdorff-closed relative, were also presented several theories related to these definitions were presented and the relationship between them was studied . We also provided other generalizations, including nano closure continuous mappings and it was also called as nano-weaklycontinuous- mappings, as well as providing a definit

The importance of topology as a tool in preference theory is what motivates this study in which we characterize topologies generating by digraphs. In this paper, we generalized the notions of rough set concepts using two topological structures generated by out (resp. in)-degree sets of vertices on general digraph. New types of topological rough sets are initiated and studied using new types of topological sets. Some properties of topological rough approximations are studied by many propositions.

In this paper introduce some generalizations of some definitions which are, closure converge to a point, closure directed toward a set, almost ω-converges to a set, almost condensation point, a set ωH-closed relative, ω-continuous functions, weakly ω-continuous functions, ω-compact functions, ω-rigid a set, almost ω-closed functions and ω-perfect functions with several results concerning them.

In the present paper, we have introduced some new definitions On D- compact topological group and D-L. compact topological group for the compactification in topological spaces and groups, we obtain some results related to D- compact topological group and D-L. compact topological group.