International Journal of Trend in Research and Development, Volume 6(3), ISSN: 2394-9333 www.ijtrd.com Understanding of Topology and the Relation between Topology and Mathematical Analysis

1Xiaomin Lei and 2,*Hongkui Li, 1,2School of Mathematics and Statistics, Shandong University of Technology, Zibo, China *Corresponding Author

Abstract - In this paper, we will study the meaning of point set A. About the open set topology and discuss the relation between topology and The open set in mathematical analysis is obtained by mathematical analysis from the aspects of open set, limit, assigning the usual topological structure to the n-dimensional connectivity, accumulation point, continuity and finite real space, and the space of an open set that satisfies a certain coverage. property is defined as the topological space. While the Keywords- Topology; Mathematical analysis; Connection 푛 dimensional real space has a typical topological structure, under which the concept of open set in mathematical analysis is I. INTRODUCTION the same as that in topology. Topology is a comprehensive discipline and is a continuation B. About limits of our previous study. The process of Euler solved the problem "Seven bridge problem" is the original "form" of topology, but In topological space, we define a sequence of convergence these problems were treated as isolated problems to deal with at as follows: 푥푖 is a sequence of topological space,and 푥 is that time. With the development of topology, these issues called a limit point of 푥푖 , if for each neighborhood of 푥,which occupy important position in the formation of topology. The called U, ∃M ∈ Z+, when푖 > 푀, 푥푖 ∈ 푈. The sequenceis called development of the topology becomes more and more mature, a convergent sequence if the sequence has at least one limit. and it is closely related to the mathematical analysis that we Obviously, in the general topological space, the limit point of have studied. Understanding these relationships will enable us the sequence is not unique, which is different from the to study these two subjects better and experience the charm of uniqueness of the sequence limit in mathematical analysis. mathematics. C. Connectivity We did not contact with topology before, so we don't know Road connectivity must be connected. Road connectivity much about it. It might seem abstract at first and is not even as and connectivity are properties that remain unchanged under easy as mathematical analysis, but in fact it has similar continuous mapping, but local connectivity is properties that properties with mathematical analysis, these properties allow us remain unchanged under continuous open mapping. They are to combine the two subjects, which can help us understand the topological invariant properties. mystery and connectedness of mathematics. Intermediate value theorem in topology is: if a continuous II. BODY mapping푓: 푎, 푏 → 푅 is a continuous mapping from the closed When we first learned topology, we felt the abstractness interval 푎, 푏 to the real number space 푅 , then for ∀푟 and profundity of it, making it difficult for people to understand. between 푓 푎 and 푓 푏 , ∃푧 ∈ 푎, 푏 makes 푓 푧 = 푟 . In However, with the continuous learning, we got more and more mathematical analysis, real number interval is connected space, in-depth understanding of it, and also found the fun of learning so it satisfies the intermediate value theorem. topology. With the further study, we gradually found that this is D. About the accumulation point not an independent subject and not as complex as we imagined, but closely related to other subjects we have learned, such as It can be seen from the mathematical analysis that there are mathematical analysis, advanced algebra, real variable function, three forms for the definition of the accumulation point: etc. (1). 푥 is a real number, and any neighborhood of푥 except point Let's take a brief look at the relationship between topology and 푥 contains the points of set A; mathematical analysis from several aspects. (2). Any neighborhood of 푥 contains endless points of Α;

IJTRD | May – June 2019 Available [email protected] 163 International Journal of Trend in Research and Development, Volume 6(3), ISSN: 2394-9333 www.ijtrd.com (3). In the set 퐴 − 푥 , there is a distinct sequence that open covering of a closed interval 푎, 푏 , then we can select a converges to 푥. finite number of open intervals from 퐻 to cover 푎, 푏 . Obviously, 푎, 푏 is a compact space. In topological space, if any neighborhood of 푥 except point 푥 contains the points of set 퐴 , then 푥 is the CONCLUSION accumulation point of 퐴, which is (1). From a topological point We have explored the connection between topology and of view, these three conditions are not true in a general mathematical analysis from the above five aspects. In addition, topological space. The (2) is valid if the topological space is 푇 , 1 there are many connections between them in other aspects. and 3 is valid if the topological space is 퐴 . 1 Topology is also closely related to other subjects, which will not E. Continuity be described here.

In mathematical analysis, we use distance to define the In the process of learning topology, we gradually realize the continuity of function, and the object of study is limited to the connection and importance between it and rich mathematical real number space. In topology, we're not just dealing with the disciplines. In the future mathematical learning, topology still real number space, but studying more general spaces. And, plays an important role, which requires us to keep learning and instead of the distance, we use the open set to define the exploring. continuity of the mapping: 푓 is a mapping from푋 to 푌 in the References topological space, and 푓 is a continuous mapping from 푋 to 푌 if the preimage of each open set 푈 in 푌 is an open set in 푋. In [1] Xiong Jincheng. Lecture Notes on Point Set Topology this way, we can study the continuous mapping of general [M]. Beijing: Higher Education Press, 2011 topological spaces in topological spaces other than metric [2] Department of Mathematics, East-China spaces. Normal University. Mathematical analysis [M] Higher Education Press, 1980 F. Finite coverage [3] Wang Yongmei, Wang Xiuyou. Deal with Extension and In point set topology, if each open covering of topological Development about Some Problems of Mathematics space 푋 has a finite subcovering, then topological space 푋 is from the View of Topology [J]. Journal of Fu Yang said to be a compact space. In mathematical analysis, if 퐻 is an Normal University, 2003(04):60-61+65

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