الفهرس 
Introduction and PreliminariesOnly 14 pages are availabe for public view
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Abstract
Mathematics is one of the oldest natural sciences inspired by mankind.
And first Arabians realized its importance where they developed geometry
and other mathematical sciences to be distinguished from the ninth century to the fifteenth century. And they have been credited for using algebra and trigonometric calculations. Mathematics has an important role in developing human mind and excel it to an advanced level. Anyone who search in mankind history realizes the importance of mathematics and its role in human brain development.
Mathematics was developed, increased in its branches and became dependent in a different knowledge areas, such as medicine, geometry and sciences humanitarian, which consider assistant base to solve many complex problems in different areas and its growth assisted the scientific progress. In recent years, importance of mathematics emerged in the era of information
revolution and technological awakening in the world. We may find that mathematics has a lot of areas, including humanitarian, political and business administration. Mathematics also played a direct role in economic development. Analysis, decision-making, planning of economic and social management is no longer possible without the ways and means of mathematics. Mathematics has contributed in the advanced areas such as information technology, launch and manufacture of satellites. The emergence and development of mathematics emerged in the boom industry
of computers, but the foregoing is a brief and quick touch to the various branches of the applications of mathematics. We are not exaggerate if we
say that we live in the age of mathematics applications.
The study of general topology is concerned with the category ”TOP”
of topological spaces as objects and continuous mappings as morphisms.
The concepts of space and mapping are equally important and one can even look at a space as a mapping from this space onto a one-point space and in this manner identify these two concepts. With this in mind, a branch of
general topology which has become known as general topology of continuous mappings or fibrewise general topology (briefly fibrewise topology), was initiated. Fibrewise topology is concerned most of all in extending the main notions and results concerning topological spaces to continuous mappings. I. M. James has been promoting the fibrewise viewpoint systematically in topology [34-39]. As a matter of fact in many directions interests in research on fibrewise theory are growing now.
Fibrewise topological spaces theory, presented in the recent 20 years, is a
new branch of mathematics developed on the basis of general topology,
algebra topology and fibrewise spaces theory. It is associated with differential geometry, Lie groups and dynamical systems theory. Therefore,
the discussion of new properties and characteristics of the variety of fibre
topological space has more important significance. To a great extend, the
research in this field has been directed to generalize to fibrewise topological spaces, notions and results classically studied in general topology. The study
of bitopological spaces was first initiated by Kelly [41] in 1963 and
thereafter a large number of papers have been done to generalize the topological concepts to bitopological setting.
Topological graph theory [6, 8, 12, 85, 86] is a branch of mathematics,
whose concepts exists not only in almost all branches of mathematics, but
also in many real life applications. We believe that topological graph structure will be an important base for narrow the gap between topology and
its applications. Also, the notions of closure operator and closure system are
very useful tools in several sections of mathematics. As an example, in
algebra [13, 14, 16], topology [24, 46, 47], computer science theory [80,
93], structural analysis [17, 18, 48], chemistry [87] and physics [30].
Topology has many subfield and lines of research, one of these directions in
topological operators which are powerful concepts that play an important
part in practical applications and also to solve many real-life problems.
The theory of rough set emerged at the end of the twentieth century. It is a mathematical model for study and analysis the overlapping data by dividing these data into equivalence classes using equivalence relations
which result from the same data [70, 71]. Applications emerged for this theory in some life fields and problems. We have seen in the past few years,
rapid growth of the world’s attention to this theory and its large applications.
Many areas of applications still in the queue waiting for the entry of the rough set theory to consider, such as the control, databases, … etc. In the last decade of the twentieth century, several workshops, global conferences and seminars dealing with the rough set mainly in their programs and was the product of these meetings, a large number of research in various branches all dealing with rough set theory. Some surveys of rough set theory and applications are presented in [67-69, 99, 108]. Some papers introduced a generalization of rough set theory [42, 52, 54, 55, 68, 69, 73, 99, 100, 101,
109] where the equivalence relations have been replaced by nonequivalence relations. Rough set theory has a close relationship with topology [42, 43, 45, 54, 72, 97, 99-101, 107, 109, 110]. Some researchers had to expand the rough sets to fuzzy sets [26, 61, 76, 93-95, 106].
Algebraic properties of fuzzy sets have been studying [21,26,32, 33, 65, 96].
The purpose of the present work is to put a starting point for the fibrewise near topology and the applications of abstract topological graph theory into rough set theory, granular computing [98] and fuzzy set theory.
Also, we will integrate some ideas in terms of concepts in topological graph
theory and uncertainty. Therefore we chose the thesis title: NEW STUDIES FOR TOPOLOGICAL GENERALIZATIONS
AND UNCERTAINTY IN GRAPH THEORY”