On some generalized metric preserving function

This thesis consists of five chapters, where the basic properties of metric preserving functions and generalized metric preserving functions are examined. Additionally, the gluing lemma for these concepts has been stated and proven. The first chapter of this thesis is the introduction, which provides a summary of the literature. In the second chapter, the fundamental concepts that will be necessary throughout the thesis are provided. In the third chapter, the concept of metric preserving function and its basic properties are examined, and two different gluing lemmas for metric preserving functions are stated and proven. In the fourth chapter, the concept of generalized metric preserving function is introduced, and the relationship between this concept known in the literature is discussed. Additionally, the gluing lemma for generalized metric preserving functions is introduced and proven. In the fifty chapter, a different circle definition based on the generalized metric preserving function is given, and examples are provided to explain the relationship between them. This final approach holds significant importance in demonstrating that the geometric interpretation of the theoretical work obtained is also possible. The sixth chapter contains conclusions and recommendations.