الفهرس 
PreliminariesOnly 14 pages are availabe for public view
Abstract
The notion of a topological group goes back to the second half of the nineteenth century. After a certain period of experimentation with the concept of a topological group it became clear that the basic thing was the continuity of the group operations. By early thirties of the twentieth century the concept of a topological group had been widely accepted. Now we find that the notion of topological group arises in almost all parts of mathematics from functional analysis to computer science as in [2, 4, 6, 18, 24, 37, 40 and 44].
It is well-known that many papers are devoted to the study of classes of subsets of a topological space, containing the class of open sets and possessing properties more or less similar to those open sets. N-Levine has introduced semi-open sets and others has introduced a collections of generalized open sets like α-open sets, feebly open sets, preopen sets and β-open sets. À.CSA ̀SZAR found that these previous kinds of generalized open sets can be deduced from suitable more general definitions; his main tool is the use of the mappings 𝛾:𝑃(𝑋) →𝑃(𝑋) from the power set 𝑃(𝑋) of the underlying set 𝑋 into itself, possessing the property of monotony (i.e. such that 𝐴⊂𝐵 implies 𝛾(𝐴)⊂𝛾(𝐵)). He denote by Γ(𝑋) the collection of all mappings having this property.
s devoted to generalize the concept of topological i thesisThis groups in generalized topological spaces instead of the topological spaces and study many basic properties of generalized topological group, generalized quotient group, generalized product of topological groups, generalized open homomorphism, generalized closed graph of the product of two generalized topological groups, GB(l)group and GB(𝒜) group, generalized topological ring and some deviations between topological groups and generalized topological groups.
contains the basic concepts and notions :chapter1 The introductory generalized sets and generalized topological spaces. It also contains the basic concepts and properties of topological groups. But, in the final section we introduced the generalized connectedness and generalized compactness in generalized topological spaces. In chapter 2 the generalized topological groups are introduced. We study the neighborhood system of any element of a generalized
group and important properties of fundamental system of generalized open neighborhoods of the identity. The separation axioms and subgroups of any generalized topological group and some deviations between topological groups and generalized topological groups are also introduced.
We organize the concepts of this chapter in four sections. In section 2.1 we generalize the notions of topological groups into generalized topological space sense; and a study of many important properties of generalized topological groups are introduced. In section 2.2 the generalized neighborhood system are introduced and properties of the generalized neighborhood system in general and specially of the fundamental system of generalized open neighborhood of the identity are shown. In section 2.3 important properties of the separation axiom in generalized topological groups are shown.
In section 2.4 we proved that a subgroup with a generalized relative topology is a generalized topological group. Also, many properties of subgroups in a generalized topological group are introduced.
the notation of generalized topological group on In chapter 3 quotient group is introduced and some of its important properties are studied. Also the notation of product of generalized topological groups is introduced and important properties of it are studied.
We organize the concepts of this chapter in two sections. In section 3.1 the generalized quotient topological group is introduced, also many notions and properties of it are studied. In section 3.2 the product of a family of generalized topological is introduced, also many notions and properties of it are studied. In chapter 4 we study many important properties of a generalized homomorphism function between two generalized topological groups, specially generalized open homomorphism. Also we study many important properties of a generalized closed graph of the product of two generalized topological groups. Finally we introduce the concepts of 𝐺𝐵(𝑙)group and 𝐺𝐵(𝒜) group and many properties of them.
We organize the concepts of this chapter in two sections. In section 4.1 we study many important properties of generalized homomorphism, almost generalized continuous, generalized continuous, almost generalized open, and generalized open functions between two generalized topological groups, Also we study many important properties of a generalized closed graph of the product of two generalized topological groups.
In section 4.2 we introduce the concepts of 𝐺𝐵(𝑙)group and 𝐺𝐵(𝒜) group and many important properties of them. In chapter 5 the generalized topological ring is introduced and basic properties of generalized topological ring are given. Neighborhood system and subrings of generalized topological ring are studied. The generalized quotient topological ring and generalized product topological ring are introduced. Also many properties of them are studied.
We organize the concepts of this chapter in four sections In section 5.1 we generalize the generalized topological group to generalized topological ring and study many properties of it. In section 5.2 we study the neighborhood system of generalized topological ring and important properties of it and we present some deviations between a topological ring and a generalized topological ring. In section 5.3 we study some properties of subrings of generalized topological rings. In section 5.4 we generate generalized topological ring on quotient ring and study important properties of it, we generate generalized topological ring on product of generalized topological spaces.
Many of the results from chapter two had published in [36] and many of the results from chapter three had published in [37]. Also the results of Section 1.4, Chapter 4 and Chapter 5 are submitted for Publication.