الفهرس 
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Abstract
This thesis, which consists of four chapters, is devoted to provide a systematic account of basic properties of what we call P-uniform spaces in order to throughout more light on the effect of the closure of the uniformity of P-uniform spaces under countable intersections instead of finite intersections only, pointing both parallelisms and vicissitudes. In addition, some applications in P-topological groups and metric spaces are studied.
The reader of this thesis is presumed to have some background in algebra and general topology, to some extend, at least of feeling at home with the basic concepts. So in Chapter (1), we give an exposition of some needed preliminaries and facts in algebra and general topology which play an important role in our study.
In Chapter (2), we introduce the concept of a P-space as a generalization of a topological space and study some basic properties and definitions. We also discuss how to determine a P-topology in terms of P-neighborhood, P-closure, and P-interior operators in a P-space, which are generalizations of operators in the topological space. Moreover, continuity in a P-space is discussed. At the end of this chapter, some properties of G Ii -sets and F a -sets in a P-space are studied.
Many authors introduced the concept of a uniform space as abstraction of a metric space. They introduced uniform spaces and obtained all basic results by extending to them the methods of theory of metric spaces. Therefore, in Chapter (3), we introduce the concept of a P-uniform space as a generalization of a uniform space and study some of its basic properties. Moreover, we show that the topology, which is induced by P-uniformity, is a P-topology. The concepts of a P-uniform sum, P-filters ... etc are given with some basic results which concern on
these concepts. At the end of this chapter, some results about completeness, completion, and LindelOf property are established.
In Chapter (4), we introduce the concept of a P-topological group and study some of its basic properties. New concepts related to a P-uniform space such as a P-unifonn group, a P-uniform subgroup, are introduced. Moreover, applications in P-topological groups and metric spaces are studied. At the end of this chapter, the concept of a quasi-P-uniform space is given with some basic results which concern on this concept.
The main results of this thesis are accepted for publication as a paper in El-Minia Science Bulletin, Volume 19(1), (ref. [3]). Other results are formulated in another paper which is submitted for publication.