الفهرس 
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Abstract
Mathematics originated and evolved to solve life problems and is still considered the language that transmits the problems of life to solutions by means of software in all areas, from the programs of mobile phones and household appliances up to the launch of satellite and intercontinental mis- siles as well as medical and engineering applications and other. Topology is one of the sciences that pepole have long thought to be far from the appli- cations of life, but at the end of the twentieth century many applications of the various branches of topology emerged. The theory of topological spaces, is the main entrance to the study of all branches of topology. So we chose the subject of our study to be on the expansions of topological spaces aimed to study the influence of expansions on compactness and on the accuracy of decision in information systems.
Compactness grew out of one of the most productive periods of math- ematical activity. In the mid to late nineteenth century, mathematicians began to really understand and specify essential properties of the real line. This work led to two different characterizations of the notion which came to be known as compactness. One characterization, developed by Bolzano and Weierstrass among others, grew out of the study of functions defined on sequences of real numbers. The other characterization, which grew out of work by Heine, Borel, and Lebesgue, was based on topological features, such as the covering of sets by open neighborhoods, [15, 35].
The rough set theory has been introduced by Zdzislaw Pawlak in 1982. It is a very satisfactory mathematical tool for representing, reasoning and de-
cision making in the case of uncertain information, [17, 28, 29]. This theory deals with the approximation of sets or concepts by means of equivalence relations and is considered as one of the first non-statistical approaches in data analysis. Several interesting applications of the theory have come up, in particular, in Artificial Intelligence and Cognitive Sciences, [2 , 37]. The main advantage of rough set theory in data analysis is that, it dose not require and preliminary or additional information of the data. The main difference between rough sets and fuzzy sets is that the rough sets have precise boundaries whereas fuzzy set theory is generally based on ill-defined sets of data, where the bounds are not precise and hence fuzzy predictions tend to deviate from exact values, [26, 27].
In 1947 G. Choquet introduced the notion of a grill with the aim of gen- eralizing the concept of a topological space, [6]. It generates finer topology which helps in investigating properties that were difficult to be deduced using the original topology. There are convenient relations between the concept of a grill and some well known concepts such as ideals, nets and filters. In [32], Roy and Mukherjee introduced and investigated the notion of the topology τg associated to a grill g on a topological space (X, τ ). During, the past twenty years, the study of continuity, compactness, nano closed sets and irresolute functions has been generalized, using the concept of a grill and many interesting constructions, properties and characteriza- tions have been deduced, cf [1, 3 , 4 ,8 , 9, 12, 13, 21, 23, 24, 36].
The main objective of this thesis is twofold. First, we aim introduced and investigate several generalizations of compactness of topological spaces
using topologies induced by grills. Actually, we introduce the notions of αg-compactness, relative αg-compactness and countably αg-compactness for a topological space (X, τ ), where g is a grill on (X, τ ). We deduce the relations between these concepts as well as the relation between these gen- eralized concepts and the original concept. Second, we aim to investigate the influence of the topology induced by a grill on the accuracy of approx- imations based on a medical information system. Roughly speaking, we prove that the accuracy of approximation has been improved. When using the topology induced by a grill.
This thesis consists of three chapters. In what follows we give a brief coverage of the contents of the thesis.
In the first chapter we assemble the basic concepts, definitions, and as- sociated well known results which are necessary for our study.
The second chapter consists of four sections. In the first section we trace the construction of the topology τg induced by a grill g on a topological space (X, τ ). The second section is devoted to the study of g-compactness,
[30] and αg-compact spaces. Actually, we prove that αg-compactness im- plies g-compact and if g is the grill consist of all non-empty subsets of a topological space (X, τ ) and the space (X, τg) is αg-compact, then (X, τ ) is α-compact. We also compare αg-compactness of (X, τ ) and (X, τg). The third section we pay attention to the study of αg-compact sets relative to a space and countably αg-compact spaces. Actually, we prove that if (X, τ )
is a space with a grill g on X and A is αg-compact relative to X, then (A, τ/A) is g/A-compact, but the converse is not true. Also, we can prove that if g is the grill consists of all non-empty subsets of (X, τ ), then the space (X, τ ) is countably α-compact if and only if (X, τ ) is countably αg- compact. We round off this chapter by establishing the relation between grill and compactness of Hausdorff spaces. Actually, we can prove that if g is a grill on a topological space (X, τ ), such that τ \ ∅ ⊆ g and (X, τ ) is αg-compact then (X, τ ) is α-quasi H-closed (α-QHC, in short). Also, we can prove that if g is a grill on a Hausdorff space (X, τ ) and (X, τ ) is αg-compact then it is αg-regular. The results of this chapter seems to be original and have been accepted for publication in:
Italian Journal of Pure and Applied Mathematics, Vol. 42.
The third chapter is divided into two sections. In the first section we provide a brief survey of the theory of rough sets and approximation. We provide examples illustrating the importance of rough sets technique ind- educing the accuracy of approximations of a given information system, cf [26, 27, 28, 29]. The second section is devoted to apply the expanded topologies induced by grills to improve the accuracy of decision of a medi- cal information system. Actually, we prove that if (U, τR) is a topological space associated with information system (U, R) and τ´ is a topology such
that τR ⊆ τ´ then Γ(A) ≤ Γ´(A), for any A ⊆ U, where Γ and Γ´
are the
accuracies with respect to τR and τ´, respectively . The main results of this chapter have been submitted for publication.