The Historical Study Of Two Important Theories In Topology

From Euclidean space to general topological space, mathematics space has taken a long time to develop. As mathematics axiomatic methods developed during the late 19th and early 20th centuries, the axiom system of topological space based on open sets was also built and had been gradually improved. By 1960s, general topological space theory has been very perfectly developed, people began seeking for a new way to study general topological space. L.A.Zadeh, a famous American cybernetician, put forward the concept of fuzzy set in 1965, and established fuzzy topological space with others (such as Chin-Liang Chang). Ying Ming Sheng from Tsinghua University defined both fuzzifying topological space and fuzzy bitopological space in 1991, thus promoted the development of the concept of topological space.In 1922, the 14 sets theorem in general topology was entitled in the article of On the Topological Closure Operation by Kazimierz Kuratowski. In 1950, Chung Tao Yang obtained C.T.Yang Theory. As two important theorems in general topological spaces, the 14 sets theoigm and C.T.Yang Theory have both differences and relations. Taking their emergence, development and historical influence as the main line, this paper clearly sorts out their development from general topological space to fuzzy topological space, and discusses the relationship between the two theorems in details.In this paper, the following methods are used:First, literature review. Through literature review, this paper respectively analyzes the 14 sets theorem and C.T.Yang Theory in generation background, development course, application and promotion, and discusses the relationship between the two, thus to enable us to understand the continuity of the two theorems'concepts and methods. Not only does this paper provide a historical reference for us to research and clarify the research direction, but also provides the basis for the further research and summarizes their historical significance and influence on the development of mathematics.Second, making the past serve the present and independent innovation. One of the significances of the study on the history of mathematics is to obtain reference and draw lessons from the development of history, and to promote real mathematical research, as commonly stated "making the past serve the present". Clearly analyzing the 14 sets theorem and C.T.Yang Theory from generation to today's development process, we have found many problems worthy of further study. Then apply some important research methods of predecessors to the study on these problems, so as to complete the mathematical innovation.Third, comparative analysis. Comparative analysis refers to the comparison between several related comparable objects, revealing their similarities and differences and drawing conclusions through analysis. The method of comparative analysis can be divided into horizontal comparison and vertical comparison. This paper horizontally compares the similarities and differences between the 14 sets theorem and C.T.Yang Theory, and vertically compared in their developments in different historical periods. We also adopt horizontal comparison in each vertical line, comparing different research results in same periods. And in horizontal comparisons, we also have horizontal comparisons of the two problems. This kind of cross comparison contributes to our comparative analysis.The main research results of this paper are as follows:First, the generation and development course of both the 14 sets theorem and C.T.Yang Theory is sorted out in a systematic way, and their historical significance and influence on the development of mathematics is discussed.Second, the similarities and differences between the 14 sets theorem and C.T.Yang Theory are studied by the method of comparative analysis, and the relationship between the two is summarized.Third, through the study of the original documents of the 14 sets theorem, we find some unsolved problems in the development process by the research method of "making the past serving the present and independent innovation". Based on learning previous practical ideas, the 14 sets theorem and some related properties in I-fuzzy topological space are put forward for the first time in this paper. At the same time, we list those problems yet to be solved, providing a clear direction for mathematical study and basis for the formulation of strategies.Fourth, the exchange situation between Kazimierz Kuratowski and the Chinese mathematics is sorted out and researched in a systematic way for the first time.Fifth, Chung Tao Yang's early work is carefully observed and studied through the method of literature review, and we provide more accurate and more detailed conclusions, and sort out his first paper.Sixth, the work of Kazimierz Kuratowski and Chung Tao Yang in elementary mathematics is sorted out and researched.Seventh, main monographs about the 14 sets theorem and C.T.Yang Theory are sorted out, and the documents of Chinese mathematicians collected in General Topology, the most important monograph in topology, are sorted out and researched.