DOCSLIB.ORG

The Importance the Importance of Topology Topology

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
The Importance the Importance of Topology Topology International Journal of Applied Science and Mathematics Volume 5, Issue 2, ISSN (Online): 2394-2894 The Importance of Topology Jing Wang Hongkui Li School of Mathematics and Statistics, Shandong School of Mathematics and Statistics, Shandong University of Technology, Zibo, China University of Technology, Zibo, China Email ID: [email protected] Email ID: [email protected] Abstract – Space is an important element of have different clustering results, which is very mathematical research. The study of spatial helpful in studying clustering algorithms of properties is a foremost task in topology. This paper machine learning[2]. At the same time, some discusses the importance of topology in three aspects. problems such as sequence convergence and fixed Firstly, taking a mathematical analysis problem as an point theorem can also be conveniently studied in example, we discuss the convenience of topology in metric space. solving other subject problems. Secondly, we analysis the applications of topological invariant in our real However, while using the metric space defined life. Finally, the spatial classification in topology by distance facilitates our research, it can also makes the topology in research problems have played hinder us in solving certain problems. For example, a transitional role. if the domain of a function is integer, is this function continuous? We know that in Keywords – Topology, Importance, Topological mathematical analysis, to solve this problem is Invariant, Spatial Classification. difficult. Then mathematical researchers will consider to solve this problem in a different space. I. INTRODUCTION The following proof that this problem is very easy to solve in topological space. Mathematics is a magical subject. We have In the topology, instead of defining distance, we different results when we define different spaces, redefine open sets, neighborhoods, closed sets, and their scope of applications are different. Metric closures and interior directly using the set as a basis space solves the problem of convergence and and define the topological space in a variety of similarity, and can prove the existence and ways. In topological space, we use open sets to uniqueness of solution using fixed point theorem. describe continuous mappings: Let X and Y be Normed linear space solves the problem of two topological spaces. f : X Y . If the elemental linear operations closure, which can preimage of each open set U in Y is an open set solve the application problem using the four major theorems (Hahn-Banach theorem, uniform bounded in X , then f is a continuous mapping from X to theorem, open mapping theorem, and closed image Y. Then find a special topological space - discrete theorem). Sobolev space can solve the weak topological space: Suppose M is a set. Let solution of differential equations[1]. However, with (M ) , that is, a family of all subsets of M. the continuous development of mathematical Then is a topology of M , , which is called a research, it is difficult for people to confine discrete topology, and we call the topology space themselves to measuring the space to study the (M , ) is a discrete space[3]. The characteristic of problem. So when people start to expand the scope of their research, there appears the discipline of discrete topological space is that every subset of it topology. Topology has its own function in our is open sets. With the above bedding, we can easily daily study life. get: 1) Let integer set Z be a subset of the real II. THE MAIN RESULTS number space (R, ) . We define the topology of Z that is Z {Z Y | Y Z } . Then From the math of elementary school to high (Z, Z ) is a subspace of topology space school, and then the higher algebra, mathematical (R, ) . analysis and analytic geometry of university, the construction of these mathematical knowledge can Arbitrarily select n Z to satisfy not do without a foundation - distance. Here, 1 1 . {n} Z (n - , n ) distance is defined by artificial axiom and we may 2 2 not define the same distance under different 1 1 (n - , n ) is an open set in R , so {n} Z . situations. A nonempty set plus the distance we 2 2 define constitutes a metric space. At the core of That is, the set of single points in Z is an open set. metric space are different metrics, that is, different From the property of the open set, we can see that metric spaces often require different distance any union of these one-point sets is also an open functions, and these different functions can set. So any subset of Z is open set. Thus Z is a describe the similarity between different elements. discrete space as one of subspace of R . In metric space, different distance functions will Copyright © 2018 IJASM, All right reserved 6 International Journal of Applied Science and Mathematics Volume 5, Issue 2, ISSN (Online): 2394-2894 2) For a mapping f from a discrete space M to philosophical concepts. This can also be [6] a topological space X . M is a discrete space, understood from the topological research ideas . so any subsets of M are open sets. And for The most important aspect of topology is 1 1 studying how different space in a continuous A X , there is f (A) M . So f (A) is mapping to maintain the same amount. In real life, a open set of M. Thus f is continuous. So the topological invariant is the property of keeping any mapping from discrete space to the object constant after continual deformation. topological space is continuous mapping. That is, a line no matter how it is still a line as long From1) and 2) shows that if a function domain is as no tie knot, and a dough no matter how knead it integer space, then this function is continuous still a dough just do not tearing it up. Euler in the mapping. settlement of konigsberg bridge problem, when he So the problem of mathematical analysis that we painted the graph without considering its size and just mentioned is solved in the topology. shape is the use of this principle. At the same time As another example, when we solve a problem, it is also a great application in architectural design. we often expand the function to the result we want, Topology in graph theory, knot theory can be and it is a problem to judge whether this function applied directly to the architectural design can be expanded. In topology, we study the Tietze modeling. Connectivity, topology embedding and expansion theorem: Let X be a topological space other characteristics are infiltrated into the building space. This has become the theoretical basis for the and be a closed interval. Then X is a [7] [a, b] development of space . Therefore, when normal space if and only if for any a closed set A discussing the properties of connectedness, axiom in X and any a continuous mapping of countable and separability, we discuss whether f : A [a, b] these properties are invariant under continuous mapping. And then study the topological invariant there is a continuous map g : X [a, b] is the properties of geometric shapes, so as to linking expansion of f [3]. According to this theorem we with the actual study of different space in a can determine whether a function can be expanded. continuous change in the amount of unchanged. For example, a function 1 can not be From another perspective, in the view of space, f x sin x topology discusses the problem of classifying space. For example, one-dimensional Euclidean expanded to a continuous map on X [a, b] . space and two-dimensional Euclidean space is not Of course, this is just a small example of using the same as a one-dimensional removal of a point topology to solve mathematical analysis problems. into two non-connected intervals, and two- Topology has many applications in other branches dimensional removal of a point is still a connected of mathematics and other subjects. In other area. The difference between a two-dimensional disciplines of mathematics, the application of European space and a three-dimensional European topological simple hematology theory to the space is that one point is removed in two formation of polyhedron embodies the process of [4] dimensions and the rubber band around this point is algebraic transformation of geometric problems ; reduced by less than one point, but three- In chemistry and chemical industry, we introduce dimensional is possible. By different topological the concept of topological index and we use properties to distinguish between different spaces, topological chemical reaction control principle in [5] so as to achieve the results we need. At the same synthetic chemistry and industrial production ; In time we can also build a fabric model based on this philosophy, the study of topology has a great fit principle. Among them, the fiber can be expressed with the nature of philosophical research. First, the in a set of real numbers. Due to different philosophical concepts discussed in the conceptual perspectives, we can divide it into fiber structure space are time-dependent. Philosophy is not space and fiber space. Both spaces have concerned with the special properties of these corresponding measures to measure the gap rheologies, but with the general nature of the between any two different fibers in space[8]. In conceptual space that holds these rheologies. This particular, we are in the study of topological space has in common with the topological study of the at the same time it is slowly moving toward the concept of space; Second, philosophy emphasizes metric space or n-dimensional European space the continuity and universality of concepts. The closer. Thus appeared Hausdorff space.
Load more

Recommended publications
  • Basic Properties of Filter Convergence Spaces
    Basic Properties of Filter Convergence Spaces Barbel¨ M. R. Stadlery, Peter F. Stadlery;z;∗ yInstitut fur¨ Theoretische Chemie, Universit¨at Wien, W¨ahringerstraße 17, A-1090 Wien, Austria zThe Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, NM 87501, USA ∗Address for corresponce Abstract. This technical report summarized facts from the basic theory of filter convergence spaces and gives detailed proofs for them. Many of the results collected here are well known for various types of spaces. We have made no attempt to find the original proofs. 1. Introduction Mathematical notions such as convergence, continuity, and separation are, at textbook level, usually associated with topological spaces. It is possible, however, to introduce them in a much more abstract way, based on axioms for convergence instead of neighborhood. This approach was explored in seminal work by Choquet [4], Hausdorff [12], Katˇetov [14], Kent [16], and others. Here we give a brief introduction to this line of reasoning. While the material is well known to specialists it does not seem to be easily accessible to non-topologists. In some cases we include proofs of elementary facts for two reasons: (i) The most basic facts are quoted without proofs in research papers, and (ii) the proofs may serve as examples to see the rather abstract formalism at work. 2. Sets and Filters Let X be a set, P(X) its power set, and H ⊆ P(X). The we define H∗ = fA ⊆ Xj(X n A) 2= Hg (1) H# = fA ⊆ Xj8Q 2 H : A \ Q =6 ;g The set systems H∗ and H# are called the conjugate and the grill of H, respectively.
    [Show full text]
  • Topology and Data
    BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 46, Number 2, April 2009, Pages 255–308 S 0273-0979(09)01249-X Article electronically published on January 29, 2009 TOPOLOGY AND DATA GUNNAR CARLSSON 1. Introduction An important feature of modern science and engineering is that data of various kinds is being produced at an unprecedented rate. This is so in part because of new experimental methods, and in part because of the increase in the availability of high powered computing technology. It is also clear that the nature of the data we are obtaining is significantly different. For example, it is now often the case that we are given data in the form of very long vectors, where all but a few of the coordinates turn out to be irrelevant to the questions of interest, and further that we don’t necessarily know which coordinates are the interesting ones. A related fact is that the data is often very high-dimensional, which severely restricts our ability to visualize it. The data obtained is also often much noisier than in the past and has more missing information (missing data). This is particularly so in the case of biological data, particularly high throughput data from microarray or other sources. Our ability to analyze this data, both in terms of quantity and the nature of the data, is clearly not keeping pace with the data being produced. In this paper, we will discuss how geometry and topology can be applied to make useful contributions to the analysis of various kinds of data.
    [Show full text]
  • A TEXTBOOK of TOPOLOGY Lltld
    SEIFERT AND THRELFALL: A TEXTBOOK OF TOPOLOGY lltld SEI FER T: 7'0PO 1.OG 1' 0 I.' 3- Dl M E N SI 0 N A I. FIRERED SPACES This is a volume in PURE AND APPLIED MATHEMATICS A Series of Monographs and Textbooks Editors: SAMUELEILENBERG AND HYMANBASS A list of recent titles in this series appears at the end of this volunie. SEIFERT AND THRELFALL: A TEXTBOOK OF TOPOLOGY H. SEIFERT and W. THRELFALL Translated by Michael A. Goldman und S E I FE R T: TOPOLOGY OF 3-DIMENSIONAL FIBERED SPACES H. SEIFERT Translated by Wolfgang Heil Edited by Joan S. Birman and Julian Eisner @ 1980 ACADEMIC PRESS A Subsidiary of Harcourr Brace Jovanovich, Publishers NEW YORK LONDON TORONTO SYDNEY SAN FRANCISCO COPYRIGHT@ 1980, BY ACADEMICPRESS, INC. ALL RIGHTS RESERVED. NO PART OF THIS PUBLICATION MAY BE REPRODUCED OR TRANSMITTED IN ANY FORM OR BY ANY MEANS, ELECTRONIC OR MECHANICAL, INCLUDING PHOTOCOPY, RECORDING, OR ANY INFORMATION STORAGE AND RETRIEVAL SYSTEM, WITHOUT PERMISSION IN WRITING FROM THE PUBLISHER. ACADEMIC PRESS, INC. 11 1 Fifth Avenue, New York. New York 10003 United Kingdom Edition published by ACADEMIC PRESS, INC. (LONDON) LTD. 24/28 Oval Road, London NWI 7DX Mit Genehmigung des Verlager B. G. Teubner, Stuttgart, veranstaltete, akin autorisierte englische Ubersetzung, der deutschen Originalausgdbe. Library of Congress Cataloging in Publication Data Seifert, Herbert, 1897- Seifert and Threlfall: A textbook of topology. Seifert: Topology of 3-dimensional fibered spaces. (Pure and applied mathematics, a series of mono- graphs and textbooks ; ) Translation of Lehrbuch der Topologic. Bibliography: p. Includes index. 1.
    [Show full text]
  • General Topology
    General Topology Tom Leinster 2014{15 Contents A Topological spaces2 A1 Review of metric spaces.......................2 A2 The definition of topological space.................8 A3 Metrics versus topologies....................... 13 A4 Continuous maps........................... 17 A5 When are two spaces homeomorphic?................ 22 A6 Topological properties........................ 26 A7 Bases................................. 28 A8 Closure and interior......................... 31 A9 Subspaces (new spaces from old, 1)................. 35 A10 Products (new spaces from old, 2)................. 39 A11 Quotients (new spaces from old, 3)................. 43 A12 Review of ChapterA......................... 48 B Compactness 51 B1 The definition of compactness.................... 51 B2 Closed bounded intervals are compact............... 55 B3 Compactness and subspaces..................... 56 B4 Compactness and products..................... 58 B5 The compact subsets of Rn ..................... 59 B6 Compactness and quotients (and images)............. 61 B7 Compact metric spaces........................ 64 C Connectedness 68 C1 The definition of connectedness................... 68 C2 Connected subsets of the real line.................. 72 C3 Path-connectedness.......................... 76 C4 Connected-components and path-components........... 80 1 Chapter A Topological spaces A1 Review of metric spaces For the lecture of Thursday, 18 September 2014 Almost everything in this section should have been covered in Honours Analysis, with the possible exception of some of the examples. For that reason, this lecture is longer than usual. Definition A1.1 Let X be a set. A metric on X is a function d: X × X ! [0; 1) with the following three properties: • d(x; y) = 0 () x = y, for x; y 2 X; • d(x; y) + d(y; z) ≥ d(x; z) for all x; y; z 2 X (triangle inequality); • d(x; y) = d(y; x) for all x; y 2 X (symmetry).
    [Show full text]
  • DEFINITIONS and THEOREMS in GENERAL TOPOLOGY 1. Basic
    DEFINITIONS AND THEOREMS IN GENERAL TOPOLOGY 1. Basic definitions. A topology on a set X is defined by a family O of subsets of X, the open sets of the topology, satisfying the axioms: (i) ; and X are in O; (ii) the intersection of finitely many sets in O is in O; (iii) arbitrary unions of sets in O are in O. Alternatively, a topology may be defined by the neighborhoods U(p) of an arbitrary point p 2 X, where p 2 U(p) and, in addition: (i) If U1;U2 are neighborhoods of p, there exists U3 neighborhood of p, such that U3 ⊂ U1 \ U2; (ii) If U is a neighborhood of p and q 2 U, there exists a neighborhood V of q so that V ⊂ U. A topology is Hausdorff if any distinct points p 6= q admit disjoint neigh- borhoods. This is almost always assumed. A set C ⊂ X is closed if its complement is open. The closure A¯ of a set A ⊂ X is the intersection of all closed sets containing X. A subset A ⊂ X is dense in X if A¯ = X. A point x 2 X is a cluster point of a subset A ⊂ X if any neighborhood of x contains a point of A distinct from x. If A0 denotes the set of cluster points, then A¯ = A [ A0: A map f : X ! Y of topological spaces is continuous at p 2 X if for any open neighborhood V ⊂ Y of f(p), there exists an open neighborhood U ⊂ X of p so that f(U) ⊂ V .
    [Show full text]
  • Combinatorial Topology
    Chapter 6 Basics of Combinatorial Topology 6.1 Simplicial and Polyhedral Complexes In order to study and manipulate complex shapes it is convenient to discretize these shapes and to view them as the union of simple building blocks glued together in a “clean fashion”. The building blocks should be simple geometric objects, for example, points, lines segments, triangles, tehrahedra and more generally simplices, or even convex polytopes. We will begin by using simplices as building blocks. The material presented in this chapter consists of the most basic notions of combinatorial topology, going back roughly to the 1900-1930 period and it is covered in nearly every algebraic topology book (certainly the “classics”). A classic text (slightly old fashion especially for the notation and terminology) is Alexandrov [1], Volume 1 and another more “modern” source is Munkres [30]. An excellent treatment from the point of view of computational geometry can be found is Boissonnat and Yvinec [8], especially Chapters 7 and 10. Another fascinating book covering a lot of the basics but devoted mostly to three-dimensional topology and geometry is Thurston [41]. Recall that a simplex is just the convex hull of a finite number of affinely independent points. We also need to define faces, the boundary, and the interior of a simplex. Definition 6.1 Let be any normed affine space, say = Em with its usual Euclidean norm. Given any n+1E affinely independentpoints a ,...,aE in , the n-simplex (or simplex) 0 n E σ defined by a0,...,an is the convex hull of the points a0,...,an,thatis,thesetofallconvex combinations λ a + + λ a ,whereλ + + λ =1andλ 0foralli,0 i n.
    [Show full text]
  • Papers on Topology Analysis Situs and Its Five Supplements
    Papers on Topology Analysis Situs and Its Five Supplements Henri Poincaré Translated by John Stillwell American Mathematical Society London Mathematical Society Papers on Topology Analysis Situs and Its Five Supplements https://doi.org/10.1090/hmath/037 • SOURCES Volume 37 Papers on Topology Analysis Situs and Its Five Supplements Henri Poincaré Translated by John Stillwell Providence, Rhode Island London, England Editorial Board American Mathematical Society London Mathematical Society Joseph W. Dauben Jeremy J. Gray Peter Duren June Barrow-Green Robin Hartshorne Tony Mann, Chair Karen Parshall, Chair Edmund Robertson 2000 Mathematics Subject Classification. Primary 01–XX, 55–XX, 57–XX. Front cover photograph courtesy of Laboratoire d’Histoire des Sciences et de Philosophie—Archives Henri Poincar´e(CNRS—Nancy-Universit´e), http://poincare.univ-nancy2.fr. For additional information and updates on this book, visit www.ams.org/bookpages/hmath-37 Library of Congress Cataloging-in-Publication Data Poincar´e, Henri, 1854–1912. Papers on topology : analysis situs and its five supplements / Henri Poincar´e ; translated by John Stillwell. p. cm. — (History of mathematics ; v. 37) Includes bibliographical references and index. ISBN 978-0-8218-5234-7 (alk. paper) 1. Algebraic topology. I. Title. QA612 .P65 2010 514.2—22 2010014958 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society.
    [Show full text]
  • Topology of Differentiable Manifolds
    TOPOLOGY OF DIFFERENTIABLE MANIFOLDS D. MART´INEZ TORRES Contents 1. Introduction 1 1.1. Topology 2 1.2. Manifolds 3 2. More definitions and basic results 5 2.1. Submanifold vs. embedding 7 2.2. The tangent bundle of a Cr-manifold, r ≥ 1. 7 2.3. Transversality and submanifolds 9 2.4. Topology with Cr-functions. 9 2.5. Manifolds with boundary 13 2.6. 1-dimensional manifolds 16 3. Function spaces 19 4. Approximations 27 5. Sard's theorem and transversality 32 5.1. Transversality 35 6. Tubular neighborhoods, homotopies and isotopies 36 6.1. Homotopies, isotopies and linearizations 38 6.2. Linearizations 39 7. Degree, intersection number and Euler characteristic 42 7.1. Orientations 42 7.2. The degree of a map 43 7.3. Intersection number and Euler characteristic 45 7.4. Vector fields 46 8. Isotopies and gluings and Morse theory 47 8.1. Gluings 48 8.2. Morse functions 49 8.3. More on k-handles and smoothings 57 9. 2 and 3 dimensional compact oriented manifolds 60 9.1. Compact, oriented surfaces 60 9.2. Compact, oriented three manifolds 64 9.3. Heegard decompositions 64 9.4. Lens spaces 65 9.5. Higher genus 66 10. Exercises 66 References 67 1. Introduction Let us say a few words about the two key concepts in the title of the course, topology and differentiable manifolds. 1 2 D. MART´INEZ TORRES 1.1. Topology. It studies topological spaces and continuous maps among them, i.e. the category TOP with objects topological spaces and arrows continuous maps.
    [Show full text]
  • Algebraic Topology
    Algebraic Topology Len Evens Rob Thompson Northwestern University City University of New York Contents Chapter 1. Introduction 5 1. Introduction 5 2. Point Set Topology, Brief Review 7 Chapter 2. Homotopy and the Fundamental Group 11 1. Homotopy 11 2. The Fundamental Group 12 3. Homotopy Equivalence 18 4. Categories and Functors 20 5. The fundamental group of S1 22 6. Some Applications 25 Chapter 3. Quotient Spaces and Covering Spaces 33 1. The Quotient Topology 33 2. Covering Spaces 40 3. Action of the Fundamental Group on Covering Spaces 44 4. Existence of Coverings and the Covering Group 48 5. Covering Groups 56 Chapter 4. Group Theory and the Seifert{Van Kampen Theorem 59 1. Some Group Theory 59 2. The Seifert{Van Kampen Theorem 66 Chapter 5. Manifolds and Surfaces 73 1. Manifolds and Surfaces 73 2. Outline of the Proof of the Classification Theorem 80 3. Some Remarks about Higher Dimensional Manifolds 83 4. An Introduction to Knot Theory 84 Chapter 6. Singular Homology 91 1. Homology, Introduction 91 2. Singular Homology 94 3. Properties of Singular Homology 100 4. The Exact Homology Sequence{ the Jill Clayburgh Lemma 109 5. Excision and Applications 116 6. Proof of the Excision Axiom 120 3 4 CONTENTS 7. Relation between π1 and H1 126 8. The Mayer-Vietoris Sequence 128 9. Some Important Applications 131 Chapter 7. Simplicial Complexes 137 1. Simplicial Complexes 137 2. Abstract Simplicial Complexes 141 3. Homology of Simplicial Complexes 143 4. The Relation of Simplicial to Singular Homology 147 5. Some Algebra. The Tensor Product 152 6.
    [Show full text]
  • Vector Calculus and the Topology of Domains in 3-Space, II Jason Cantarella, Dennis Deturck and Herman Gluck
    Vector Calculus and the Topology of Domains in 3-Space, II Jason Cantarella, Dennis DeTurck and Herman Gluck 1. Introduction Suppose you have a vector field defined only on the boundary of a compact domain in 3-space. How can you tell whether your vector field can be extended throughout the domain so that it is divergence-free? Or curl-free? Or both? Or have specified divergence? Or specified curl? Or both? To answer these questions, you need to understand the relationship between the calculus of vector fields and the topology of their domains of definition. The Hodge Decomposition Theorem, which was discussed and proved in the first paper of this two-part series [12], provides the key by decomposing the space of vector fields on the domain into five mutually orthogonal subspaces which are topologically and an- alytically meaningful. Our goal in this paper is to show how this theorem can be used to derive the basic boundary-value theorems for vector fields, which in turn will answer the questions above. As we shall see, each of these questions can be restated as a problem in partial differential equations, and it will be essential for us to identify appropriate boundary conditions so that a solution will exist and be unique, or so that the set of solutions is parametrized in a meaningful way. In particular, we will consider the problem of finding an unknown vector field V on Ω whose curl and/or divergence are specified in advance. If just one of these is given, then the value of V on the boundary of Ω is also specified.
    [Show full text]
  • Geometry and Topology SHP Spring ’17
    Geometry and Topology SHP Spring '17 Henry Liu Last updated: April 29, 2017 Contents 0 About Mathematics 3 1 Surfaces 4 1.1 Examples of surfaces . 4 1.2 Equivalence of surfaces . 5 1.3 Distinguishing between surfaces . 7 1.4 Planar Models . 9 1.5 Cutting and Pasting . 11 1.6 Orientability . 14 1.7 Word representations . 16 1.8 Classification of surfaces . 19 2 Manifolds 20 2.1 Examples of manifolds . 21 2.2 Constructing new manifolds . 23 2.3 Distinguishing between manifolds . 25 2.4 Homotopy and homotopy groups . 27 2.5 Classification of manifolds . 30 3 Differential Topology 31 3.1 Topological degree . 33 3.2 Brouwer's fixed point theorem . 34 3.3 Application: Nash's equilibrium theorem . 36 3.4 Borsuk{Ulam theorem . 39 3.5 Poincar´e{Hopftheorem . 41 4 Geometry 44 4.1 Geodesics . 45 4.2 Gaussian curvature . 47 4.3 Gauss{Bonnet theorem . 51 1 5 Knot theory 54 5.1 Invariants and Reidemeister moves . 55 5.2 The Jones polynomial . 57 5.3 Chiral knots and crossing number . 60 A Appendix: Group theory 61 2 0 About Mathematics The majority of this course aims to introduce all of you to the many wonderfully interesting ideas underlying the study of geometry and topology in modern mathematics. But along the way, I want to also give you a sense of what \being a mathematician" and \doing mathematics" is really about. There are three main qualities that distinguish professional mathematicians: 1. the ability to correctly understand, make, and prove precise, unambiguous logical statements (often using a lot of jargon); 2.
    [Show full text]
  • On a New Convergence in Topological Spaces Received March 5, 2019; Accepted October 28, 2019
    Open Math. 2019; 17:1716–1723 Open Mathematics Research Article Xiao-jun Ruan and Xiao-quan Xu* On a new convergence in topological spaces https://doi.org/10.1515/math-2019-0123 Received March 5, 2019; accepted October 28, 2019 Abstract: In this paper, we introduce a new way-below relation in T0 topological spaces based on cuts and give the concepts of SI2-continuous spaces and weakly irreducible topologies. It is proved that a space is SI2-continuous if and only if its weakly irreducible topology is completely distributive under inclusion order. Finally, we introduce the concept of D-convergence and show that a space is SI2-continuous if and only if its D-convergence with respect to the topology τSI2 (X) is topological. In general, a space is SI-continuous if and only if its D-convergence with respect to the topology τSI(X) is topological. Keywords: s2-continuous poset, weakly irreducible topology, SI2-continuous space, D-convergence MSC: 06B35, 06B75, 54F05 1 Introduction Domain theory which arose from computer science and logic, started as an outgrowth of theories of order. Rapidly progress in this domain required many materials on topologies (see [1–3]). Conversely, it is well known that given a topological space one can also dene order structures (see [3–6]). At the 6th International Symposium in Domain Theory, J.D. Lawson emphasized the need to develop the core of domain theory directly in T0 topological spaces instead of posets. Moreover, it was pointed out that several results in domain theory can be lifted from the context of posets to T0 topological spaces (see [5, 6]).
    [Show full text]