Topolojik Uzaylarda Yakınlık

Abstract

This paper consists of four chapters. The fi rst chapter is an introduction which contains the basic motivation of nearness theory. The second section is devoted to nearness in metric spaces. Here, the nearness of two sets is de fined by gap functional. In particular, the closure point of a set is defi ned using nearness. The concepts of convergence of a sequence, and continuity of a function are characterized in terms of nearness. The interior of a set is also defi ned using nearness. Proximal neighbourhood of a set in a metric space is defi ned and the basic properties are discussed. Compatible proximity and fine proximity are defi ned and it is proved that every fine proximity is also metric proximity. For compact spaces, it is shown that every metric proximity is a fine proximity. The proximal continuity is defi ned and it is proved that every proximal continuous function is also continuous. The nearness in the sense of Herrlich is given and the basic properties of Herrlich nearness are presented. Using metric proximity, a characterization is given for Cauchy sequences. It is proved that uniform continuity is equivalent to proximal continuity. Further, Hausdorff metric is de fined and it is proved that every convergent closed set sequence is uniform convergent. Finally, for continuous extensions, the Taimanov Theorem is proved. In the third chapter, Efremovic proximity is de fined as a generalization of metric proximity. Then the Lodato proximity is presented and it is shown that every Efremovic proximity is a Lodato proximity. Further, compatible proximity is considered in topological spaces. In this respect, proximity is studied under certain separation properties of topological spaces. In particular, for completely regular and normal spaces, the existence of compitable proximity is discussed. In the fourth chapter, descriptive proximity is discussed in the sense of Efremovic and Lodato. Finally, spatial and descriptive proximities are compared.