Some Properties of Bitopological Ordered Spaces
Mona Hosny Abd El Khalek Ali;
Abstract
This thesis consists of seven chapters distributed as follows:
Chapter 1: is the introductory chapter and contains the basic concepts and
properties of relations and rough sets theory. It contains also the basic concepts
and properties of topological ordered, bitopological, bitopological ordered spaces.
Additionally, in Subsection 1.5.1, we introduce the operator A_12 and study somebitopological properties via ideals, since this part is used in sequel.
The object of Chapter 2 is to use the supra topological ordered space (X;_12;R)to study the bitopological ordered space (X; _1; _2;R), since the dealing with one family is easier than the dealing with two families. The class of all supra topological ordered spaces is wider than the class of topological ordered spaces. Moreover, some order separation axioms on the spaces (X; _1; _2;R) are introduced by using the properties of the space (X; _12;R). Furthermore, some important results and
some examples related to these separations have been obtained.
In Chapter 3, the notions of increasing (decreasing) sets [60] are generalized by using the concepts of the ideal. The current notions is denoted by I-increasing (I-decreasing) sets. The main properties of the present notions are studied and compared to previous one [60] and shown to be more general. Moreover, it is proved that the class of all I-increasing (I-decreasing) sets forms a topology on X and this topology is finer than the topology that is generated by increasing (decreasing) sets. Furthermore, it is proved that the topology of increasing (decreasing) sets which is generated by ideal coincides with the topology of I-increasing (I-decreasing) sets. Additionally, the ideal bitopological ordered space (X;_1; _2;R; I) is introduced as a generalization of the study of bitopological ordered space (X; _1; _2;R) and bitopological space (X; _1; _2). The separation axioms, IPTi-ordered spaces, i = 0; 1; 2; 3; 4; 5 and IPRj-ordered spaces, j = 0; 1; 2; 3; 4 are presented, their properties are discussed and compared to the separation axioms in [2, 76]. The important of the current study is that the new spaces are more general because the old one [2, 76] can be obtained from the current spaces when I = f_g: Some types of mappings which preserve the properties of the present separation axioms are introduced. At the end of this chapter the ideal supra topological ordered spaces (X; _12;R; I) are introduced and some order separation axioms namely IP_Ti- ordered spaces, i = 0; 1; 2 are studied in the ideal bitopological ordered spaces. Comparisons between all types of order separation axioms in this chapter and the previous one [76] are presented.
The goal of Chapter 4 is to study the notion of connectedness in (ideal) bitopological ordered spaces. The notions of P-(P-_)-connectedness and P_-(P_-_)-
connected ordered spaces are introduced in (ideal) bitopological ordered spaces.
Some important results related to these notions have been obtained and the relationships between the current study and the previous one [19, 50, 68, 74] have
been presented.
The main purpose of Chapter 5 is to propose different methods of generalization
rough sets theory by using topologies, filters, increasing and I-increasing sets. The
main properties of the current methods are studied and compared to El-Shafei et
al.'s approximations [18]. The important of the current results is not only that it
consider attributes with preference-ordered domains, but also it is reducing the
boundary region and increasing the accuracy of sets which is the main aim of
rough set.
The aim of Chapter 6 is to initiate a new (ideal) topological ordered approach
to generalize rough sets. The present approach connects rough sets, (ideal) topological
ordered spaces, and binary relations. It is depended on the topologies
which are generated by after sets and after-fore sets. Additionally, it is based on
the increasing and I-increasing sets to define new notions of the lower and upper
approximations. The properties of the new notions of lower (upper) approximations
are studied and some examples are used to illustrate the present concepts
in a friendly way. Comparisons between the current methods in this chapter, the
methods in Chapter 5 and El-Shafei et al.'s method [18] are presented.
Finally, in Chapter 7 the concept of rough membership is generalized by using
different methods. The important of these methods is that it consider attributes
with preference-ordered domains. It is based on the increasing and I-increasing
sets and while the first methods are depended on a subbase for a topology, the
second methods are depended on a subbase for a filter. The properties of the
suggested concepts are studied. The relationships between the suggested methods
are obtained.
Chapter 1: is the introductory chapter and contains the basic concepts and
properties of relations and rough sets theory. It contains also the basic concepts
and properties of topological ordered, bitopological, bitopological ordered spaces.
Additionally, in Subsection 1.5.1, we introduce the operator A_12 and study somebitopological properties via ideals, since this part is used in sequel.
The object of Chapter 2 is to use the supra topological ordered space (X;_12;R)to study the bitopological ordered space (X; _1; _2;R), since the dealing with one family is easier than the dealing with two families. The class of all supra topological ordered spaces is wider than the class of topological ordered spaces. Moreover, some order separation axioms on the spaces (X; _1; _2;R) are introduced by using the properties of the space (X; _12;R). Furthermore, some important results and
some examples related to these separations have been obtained.
In Chapter 3, the notions of increasing (decreasing) sets [60] are generalized by using the concepts of the ideal. The current notions is denoted by I-increasing (I-decreasing) sets. The main properties of the present notions are studied and compared to previous one [60] and shown to be more general. Moreover, it is proved that the class of all I-increasing (I-decreasing) sets forms a topology on X and this topology is finer than the topology that is generated by increasing (decreasing) sets. Furthermore, it is proved that the topology of increasing (decreasing) sets which is generated by ideal coincides with the topology of I-increasing (I-decreasing) sets. Additionally, the ideal bitopological ordered space (X;_1; _2;R; I) is introduced as a generalization of the study of bitopological ordered space (X; _1; _2;R) and bitopological space (X; _1; _2). The separation axioms, IPTi-ordered spaces, i = 0; 1; 2; 3; 4; 5 and IPRj-ordered spaces, j = 0; 1; 2; 3; 4 are presented, their properties are discussed and compared to the separation axioms in [2, 76]. The important of the current study is that the new spaces are more general because the old one [2, 76] can be obtained from the current spaces when I = f_g: Some types of mappings which preserve the properties of the present separation axioms are introduced. At the end of this chapter the ideal supra topological ordered spaces (X; _12;R; I) are introduced and some order separation axioms namely IP_Ti- ordered spaces, i = 0; 1; 2 are studied in the ideal bitopological ordered spaces. Comparisons between all types of order separation axioms in this chapter and the previous one [76] are presented.
The goal of Chapter 4 is to study the notion of connectedness in (ideal) bitopological ordered spaces. The notions of P-(P-_)-connectedness and P_-(P_-_)-
connected ordered spaces are introduced in (ideal) bitopological ordered spaces.
Some important results related to these notions have been obtained and the relationships between the current study and the previous one [19, 50, 68, 74] have
been presented.
The main purpose of Chapter 5 is to propose different methods of generalization
rough sets theory by using topologies, filters, increasing and I-increasing sets. The
main properties of the current methods are studied and compared to El-Shafei et
al.'s approximations [18]. The important of the current results is not only that it
consider attributes with preference-ordered domains, but also it is reducing the
boundary region and increasing the accuracy of sets which is the main aim of
rough set.
The aim of Chapter 6 is to initiate a new (ideal) topological ordered approach
to generalize rough sets. The present approach connects rough sets, (ideal) topological
ordered spaces, and binary relations. It is depended on the topologies
which are generated by after sets and after-fore sets. Additionally, it is based on
the increasing and I-increasing sets to define new notions of the lower and upper
approximations. The properties of the new notions of lower (upper) approximations
are studied and some examples are used to illustrate the present concepts
in a friendly way. Comparisons between the current methods in this chapter, the
methods in Chapter 5 and El-Shafei et al.'s method [18] are presented.
Finally, in Chapter 7 the concept of rough membership is generalized by using
different methods. The important of these methods is that it consider attributes
with preference-ordered domains. It is based on the increasing and I-increasing
sets and while the first methods are depended on a subbase for a topology, the
second methods are depended on a subbase for a filter. The properties of the
suggested concepts are studied. The relationships between the suggested methods
are obtained.
Other data
| Title | Some Properties of Bitopological Ordered Spaces | Other Titles | بعض خصائص الفراغات ثنائية التوبولوجي المرتبة | Authors | Mona Hosny Abd El Khalek Ali | Issue Date | 2015 |
Recommend this item
Similar Items from Core Recommender Database
Items in Ain Shams Scholar are protected by copyright, with all rights reserved, unless otherwise indicated.