The Creation Of Point Set Topology

Point set topology is a branch of mathematics which studies the spatial structure related to topology and probes into the properties of the mapping defined on the basis of the space. Not only does it have a close relation with many branches of mathematics, but also it has a variety of applications. Therefore, the study of the history of point set topology has a very important theoretical value and practical significance. Based on an extensive investigation of original sources and research papers on this subject, and using "Why mathematics was done" as the starting point and main purpose, the dissertation studies systematically the origin and invention of the point set topology by means of historical analysis and literature review. The combination of set theory, analysis and the axiomatic method is one of the features of this study. The main contributions of this study are as follows:1. Through the discussion of the causes of Cantor’s ideas, his early works in set theory and a systematic research about his two important works--"Foundations of General Set Theory" and "Contributions to the Founding of the Theory of Transfinite Numbers", the initial presentations of some important concepts and theorems on point set topology are given.2. By thoroughly discussing some specific problems in analysis which are related to the point set topology before Frechet introducing the metric space, the paper studies the analysis origin of point set topology during its invention. The contents mainly include the Weierstrass’ works on Arithmetization of Analysis, the process of Riemann’s introduction of the manifold and its impact on point set topology, some works of a lot of mathematicians, including Volterra, Ascoli, Arzela and Borel, on the early expansion of Cantor’s set theory.3. Some intensive investigations about Frechet’s contribution to point set topology are conducted, and his general theory about metric space is studied in detail in this paper, which includes, the six papers of his early study which have been neglected and have a close relation with his doctoral thesis, some deep researches about his doctorial thesis, and the analysis of his introduction about the metric space. This paper points out that, besides a systematic summary of his early related works, his doctoral thesis also contains a number of groundbreaking work. In addition, through analyzing the Frechet’s ideas, this study reaches a conclusion that the great success of his doctoral thesis is mainly due to his conformity to the trends of mathematics in the20th century, namely pursuing the "generalization" and "unity"4. By elaborating a lot of related works of some mathematicians, which include, Hilbert’s work on Hilbert space and the foundation of geometry, Riesz’s discussion about some abstract space, some related works of Weyl on Riemann surface, and the Youngs’ study about point set theory in real line and real plane, the background of the birth of point set topology is investigated, and the impact of the works of these pioneers on point set topology is analyzed. In addition, it also clarifies some issues in the history of mathematics.5. The great change and development made by Hausdorff on point set topology is analyzed. Focusing on the book "Foundation of Set Theory" published in1914, this paper points out how Hausdorff developed Hilbert’s work on plane geometry and Weyl’s discussion on Riemann surface with axiomatic methods, and then described the concept of topology space through the language of neighborhood. Meanwhile, this paper discusses how Hausdorff established a perfect theory, and explains, to what extent, his work provided a strong impetus in the development of point set topology.6. A systematic examination about the works of the related mathematicians in the formation period is conducted in this paper. Through comparing the definitions of topology space which were presented by different mathematicians, the paper points out the degree of acceptance and the manners they treated this concept in the birth time of point set topology, and clarifies some problems in the history. In addition, it discusses some studies about topological invariants, and introduces some hot issues at that time, such as the problem of metrizable topological space. Finally, it clarifies the evolution process of some concepts and ideas on point set topology.