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MA4266 Topology Lecture 1. Wayne Lawton Department of Mathematics S17-08-17, 65162749 matwml@nus.edu.sg http://www.math.nus.edu.sg/~matwml/ http://arxiv.org/find/math/1/au:+Lawton_W/0/1/0/all/0/1 Administrative MA4266 Module Title TOPOLOGY Semester 2, 2010/2011 Modular Credits 4 Faculty Science Department Mathematics Teaching Staff ASSOC PROF Lawton, Wayne Michael matwml@nus.edu.sg http://www.math.nus.edu.sg/~matwml/courses/ my personal website which contains mountains of materials AIMS & OBJECTIVES The objective of this module is to give a thorough introduction to the topics of point-set topology with applications to analysis and geometry. Major topics: topological spaces, continuous maps, bases, subbases, homeomorphisms, subspaces, sum and product topologies, quotient spaces and identification maps, orbit spaces, separation axioms, compact spaces, Tychonoff's theorem, Heine-Borel theorem, compactness in metric space, sequential compactness, connected and path-connected spaces, components, locally compact spaces, function spaces and the compact-open topology. PREREQUISITES MA3209 Mathematical Analysis III TEACHING MODES Lectures, questions, discussions, tutorial problem solving and presentation by students and subsequent discussion encouraged by questions from the lecturer, assigned readings covering most chapters from the textbook Principles of Topology by Fred Croom and supplementary materials taken from Various sources, two tests and a final examination, written homework that is collected and graded and handed back to students. SCHEDULE Final Examination LECTURE Class [SL1] TUESDAY From 1600 hrs to 1800 hrs in S16-0430, Week(s): EVERY WEEK. FRIDAY From 1600 hrs to 1800 hrs in S16-0430, Week(s): EVERY WEEK. TUTORIAL Class [T01] WEDNESDAY From 1100 hrs to 1200 hrs in S16-0433, Week(s): EVERY WEEK. SYNOPSIS described in the preface of the required textbook Principles of Topology by Fred Croom PRACTICAL WORK none ASSESSMENT Test 1 20% Test 2 20% Homework/Tutorials 20% Final Examination 40% PRE-CLUSIONS MA3251, MA4215, FASS students from 2003-2006 cohort who major in Mathematics (for breadth requirement). WORKLOADTop3-1-0-0-6 TEXT & READINGS Principles of Topology Author: Fred H. Croom, -Compulsory Textbook Principles of Topology by Fred H Croom, Thompson, Singapore, 1989. Available in the Science COOP Bookstore at a significantly reduced student price The use of this textbook is compulsory because you are expected to read most of it and work out solutions to selected problems located at the ends of each of the 8 chapters. Contents of Textbook 1. Introduction 2. The Line and the Plane 3. Metric Spaces 4. Topological Spaces 5. Connectedness 6. Compactness 7. Product and Quotient Spaces 8. Separation Properties and Metrization 9. The Fundamental Group My aim to is cover all of the material in the textbook What is Topology ? Greek “position” or “location” in the sense of properties that are NOT destroyed by continuous transformations bending, shrinking, stretching and twisting BUT are destroyed by discontinuous transformations cutting, tearing, and puncturing Example 1.1.1 Shrinking in the vertical direction ( x, y) ( x, y / 2) Non Morphable Example Involution in a circle z 1/ z on C \ {0} ( x, y ) ( x / r , y / r ), r x y 2 2 Topological Equivalence Between Geometric Objects X and Y means there exists is a continuous bijection f : X Y 1 whose inverse f :Y X is also continuous. Which pairs below of geom. obj. are top. equiv.? (0,1), R [0,1), R [0,1), (,1] Who Needs Topology ? Theorem 1.1: The Intermediate Value Theorem. If f : R R is continuous and there exists a, b R such that f (a) 0 and f (b) 0 then there exists c R such that f (c) 0. Corollary If P : R R is defined by a polynomial having odd degree then P has a real root. In what other areas, aside from calculus and algebra, are existence theorems important? Origins of Topology In 1676 Leibnitz used the term “geometria situs”, Latin for “geometry of position”, to designate what he predicted to be the development of a new type of geometry similar to modern day topology http://en.wikipedia.org/wiki/Gottfried_Leibniz Example 1.2.1 In 1736 Euler solved The Königsberg bridges problem, this inventing graph theory http://en.wikipedia.org/wiki/Seven_Bridges_of_K%C3%B6nigsberg Psychologists hypothesize that the human brain is topologically wired ! http://www.jstor.org/pss/748762 Assignment 1 Read Preface and pages 1-14 in Chapter 1 Do Problems 1-6 on pages 13-14 and prepare to solve on the board in class for Friday

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