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

Fuzzy orbit topological space is a new structure very recently given by [1]. This new space is based on the notion of open fuzzy orbit sets. The aim of this paper is to provide applications of open fuzzy orbit sets. We introduce the notions of fuzzy orbit irresolute mappings and fuzzy orbit open (resp. irresolute open) mappings and studied some of their properties. .

A new class of generalized open sets in a topological space, called G-open sets, is introduced and studied. This class contains all semi-open, preopen, b-open and semi-preopen sets. It is proved that the topology generated by G-open sets contains the topology generated by preopen,b-open and semi-preopen sets respectively.

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

  The aim of this paper is to introduce a new type of proper mappings called semi-p-proper mapping by using semi-p-open sets, which is weaker than the proper mapping. Some properties and characterizations of this type of mappings are given.

Csaszar introduced the concept of generalized topological space and a new open set in a generalized topological space called -preopen in 2002 and 2005, respectively. Definitions of -preinterior and -preclosuer were given. Successively, several studies have appeared to give many generalizations for an open set. The object of our paper is to give a new type of generalization of an open set in a generalized topological space called -semi-p-open set. We present the definition of this set with its equivalent. We give definitions of -semi-p-interior and -semi-p-closure of a set and discuss their properties. Also the properties of -preinterior and -preclosuer are discussed. In addition, we give a new type of continuous function

            This work, introduces some concepts in bitopological spaces, which are nm-j-ω-converges to a subset, nm-j-ω-directed toward a set, nm-j-ω-closed mappings, nm-j-ω-rigid set, and nm-j-ω-continuous mappings. The mainline idea in this paper is nm-j-ω-perfect mappings in bitopological spaces such that n = 1,2  and m =1,2 n ≠ m. Characterizations concerning these concepts and several theorems are studied, where j = q , δ, a , pre, b, b.

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 structure of this paper includes an introduction to the definition of the nano topological space, which was defined by M. L. Thivagar, who defined the lower approximation of G and the upper approximation of G, as well as defined the boundary region of G and some other important definitions that were mentioned in this paper with giving some theories on this subject. Some examples of defining nano perfect mappings are presented along with some basic theories. Also, some basic definitions were presented that form the focus of this paper, including the definition of nano  pseudometrizable space, the definition of nano compactly generated space, and the definition of completely nano para-compact. In this paper, we presented images of nan