الفهرس 
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Abstract
For a long time, the topology scientists have been faced
many of the questions about the importance of this science,
which ranked among the basics of pure mathematics and
seems far from the areas of application process. Sometimes
these questions were directed from specialists in the _eld
of mathematics and many non-specialists. Now it becomes
easy to answer those questions, especially that there are
some recent trends to consider the topological structure as
the basis or a knowledge based on a set of data drawn from
the experiences of life, and this helps to extract new char-
acteristics of a grouped data and in the information age,
these directions helped researchers in various aspects of
life; in the establishment of mathematical models for clus-
ters which were di_cult to deal with mathematically. Thus
the topological structures have become of great importance
in the expansion of some theories of uncertainty as rough
set theory, fuzzy set theory and probability theory.
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Summary
The topic of fuzzy information granulation was _rst pro-
posed by Professor L.A. Zadeh [85]. Based on fuzzy set the-
ory, he proposed a general framework of granular comput-
ing. Granules which are constructed and de_ned are based
on the concept of equivalence, probability and fuzzy rela-
tionship. Relationships between granules are represented
in terms of fuzzy graphs or fuzzy if-then rules. Its compu-
tation method is computing with words [63].
Rough set theory proposed by Professor Z. Pawlak in
1982 has been applied to many _elds [44, 46]. It is a valid
mathematical method to deal with imprecise, uncertain,
and vague information [53, 38, 59], as a concrete theory of
granular computing. Rough set theory enables us to pre-
cisely de_ne and analyze many notions of granular com-
puting [32].
Granular computing is a valid way to describe problem
spaces or to solve problems. It enables us to perceive
the real world under various grain sizes, obtain only those
useful or interesting things at di_erent granulations, and
switch among di_erent granulations to get various levels
of knowledge. The basic ideas of granular computing have
already been explored in many _elds, such as soft com-
puting, knowledge discovery, machine learning, and web
intelligence. At present, there are several major research
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Summary
results shown as follows.
Granular modeling (Granular computing) is a general-
ization of fuzzy set theory, rough set theory and is a label
of theories, methodologies, techniques, and tools that make
use of granules, i.e., groups, classes, or clusters of a universe
in the process of problem solving. There are many reasons
for the study of granular computing [27, 54]. The practi-
cal necessity and simplicity in problem solving are perhaps
some of the main reasons. When a problem involves incom-
plete, uncertain, or vague information, it may be di_cult to
di_erentiate distinct elements and one is forced to consider
granules. Although detailed information may be available,
it may be su_cient to use granules in order to have an ef-
_cient and practical solution. Very precise solutions may
not be required for many practical problems. The use of
granules generally leads to simpli_cation of practical prob-
lems. The acquisition of precise information may be too
costly, and coarse-grained information reduces cost. There
is clearly a need for the systematic studies of granular com-
puting.
Professors Y.Y. Yao and Q. Liu study the partition model
and logical reasoning based on granular computing [30, 66,
67, 68, 69, 70, 71]. Construction of granules and the rep-
resentation of their relationships are described in terms of
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Summary
the decision logic language [72].
Professor T.Y. Lin has contributed a lot of work to gran-
ular computing and its applications [24, 23, 25], including
granular computing based on binary relations in data min-
ing and neighborhood systems. Some basic ideas of granu-
lar computing have been successfully explored with regard
to image processing and machine learning in his work.
Professors L. Zhang and B. Zhang developed the quotient
space theory in 1992 [88, 89], it is a novel theory of gran-
ular computing. Granules are constructed and de_ned by
collection. Relationships between granules are represented
in terms of collection calculation. An important property
of quotient space theory is the ”false preserving” property,
it means that there will be no solution in the original _ne-
grained space if there is no solution in the coarse-grained
space. The quotient space theory has been applied to wave
analyzing and bioinformation processing.
Although there are di_erences among these theories of
granular computing in their formulation, i.e., construction
of granules and representation of their relationships, their
core idea is the same. That is, a problem space is _rstly di-
vided into some basic granules. Then, these basic granules
are further composed or decomposed into new granules at
di_erent hierarchies. The above two steps are repeated un-
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Summary
til these new granules could solve the problem more valid.
Since each theory has its advantages of solving the prob-
lem in some _eld, it is very important to take advantages
of each theory to solve the problem in a more e_cient way.
This thesis includes _ve chapters:
The _rst chapter is devoted to answer on the following
questions:
1) What is granular computing ?
2) Why do we study granular computing?
3) What is new and di_erent in granular computing?
4) What are basic issues of granular computing?
Furthermore, a brief survey on Fuzzy set theory and the
classical rough set theory is presented. Also, Fundamental
notions of topological spaces are given.
The second chapter is concerned with introducing and
studying the generalizations of rough set theory based on
coverings. Particularly, we de_ne the approximation space
for n- coverings de_ned on the universe of discourse. The
de_nition of topologized covering approximation space is
introduced. We generalize the approximation space by us-
ing n- coverings. We de_ne rough equality, rough inclusion
of sets, the accuracy measure and membership function
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Summary
based on n- coverings. Finally, we introduce the applica-
tion of n-coverings on information systems. Some results
of this chapter are published in:
- Journal of Institute of Mathematics and Com-
puter Sciences.
- Proceeding of 7th International Engineering Con-
ference which hold in Faculty of Engineering- Man-
soura University, March, 2010.
The third chapter deals the application of generalized
covering-based rough set theory. We analyze nondetermin-
istic information systems by covering approximation space.
We give the de_nition of the deterministic information sys-
tem. Mathematical models in deterministic information
system: indiscernibility relation, discernibility matrix and
discernibility function are presented. Moreover, we give the
meaning of nondeterministic information systems, types of
nondeterministic information systems and the reduction of
multivalued information systems. Covering approach for
multivalued information systems is introduced. Finally,
Reduction of attributes and reduction of attribute-values
are established. Some results of this chapter are published
in:
- International Journal of Applied Mathematics.
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Summary
The primary objective of The fourth chapter is the
study of the relationship between a type of covering-based
rough sets and the generalized rough sets based on binary
relation-based a type of neighborhood operator. We de-
_ne the neighborhood operator induced by a binary rela-
tion. Rough approximations based on neighborhood op-
erator are introduced. We study rough sets generated by
two relations. The _rst, we give a new type of covering-
based rough sets. The second, we establish the relationship
between neighborhood operators and covering-based rough
sets. Finally, Covering-based rough set induced by mixed
neighborhood operator is presented . Some results of this
chapter are:
- Presented in The 24rd conference of Topology and
its Applications which hold in Faculty of Sceince,
Beni-Suef University, April, 21, (2010).
- Submitted for publication in International Journal of
Computer Mathematics.
In the last chapter, we study covering-based rough
fuzzy sets in which a fuzzy set can be approximated by the
intersection of some elements in the covering of the uni-
verse of discourse. The di_erence between the concepts of
rough fuzzy set and fuzzy rough set is given. We introduce
new concepts and properties of covering-based rough fuzzy
set. The conditions under which two coverings generate
vii
Summary
the same covering-based fuzzy lower and fuzzy upper ap-
proximations are established . In addition, we approximate
fuzzy sets based on a binary relation and their properties
are introduced. Finally, the relationship between covering-
based fuzzy approximation and binary relation-based fuzzy
approximation are establishe