Introduction to Metric and Topological Spaces
Math 445 -- Fall 2009 -- Call #13713 undergrad, #20457 grad
2 PM, MWF, 306 Addams Hall
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INSTRUCTOR: Steven Hurder |
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COURSE DESCRIPTION
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This course will give an introduction to topology, which is the study of the "shape" of spaces. The subject is a mix of the study of the menagerie of examples of sets and spaces whose properties are revealed by the axioms of the subject, and careful reasoning from the axioms. The goal of the course is actually two-fold: one is to master the basics of the subject; the other is to sharpen your skills of mathematical thinking, especially using geometric intutition to discover the many new ideas of the subject. The subject of topology developed in the nineteenth century, as the need to understand the foundations of analysis and set theory became evident. Georg Cantor and Felix Hausdorff are considered the fathers of "point-set topology". Topology, as the study of the "shape of space", evolved concurrently from the study of the classical solids to the incredible richness of Riemannian geometry, especially through the works of Bernhard Riemann and Henri Poincaré. This course will emphasize the fundamentals of the subject, starting with "elementary set theory" and working up to all aspects of point-set topology in metric spaces. This textbook for the course by Irving Kaplansky is a classic text based on M.I.T. lecture notes from the 1950's. The writing in the text has great clarity, and the author In addition to point-set, the course will also discuss general ideas of topological spaces, including the Hausdorff speration axioms, with examples drawn from number theory, function spaces, and set theory. This course gives a firm foundation for further study of Algebraic Topology, Differential Topology, Real Analysis, and Complex Analysis. |
EXAMS
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Midterm Exam - Due October 26 - Solutions Final Exam - December 9 from 1-3 PM in Room 306 AH - Details to be posted |
HOMEWORK & GRADES
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Working problems is an integral part of learning this subject. Homework will be assigned each week and collected. The Final grade will be based upon grades for the homeworks, Midterm and Final Exam.
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SUPPLEMENTAL HANDOUTS
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Supplemental readings and articles will be distributed in class from time to time, to augment the textbook by Kaplansky.
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TEXTBOOK & READINGS
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The course textbook is: Set Theory and Metric Spaces, by Irving Kaplansky, American Mathematical Society; 2nd edition (May 1, 2001)There are many texts on the subject of Topology, which develop the some topics in more depth and breadth. They are strictly supplemental. But if you find one of these books used, they will make a nice addition to your library, and give more insights into the subject. Introduction to Metric and Topological Spaces, by W. A. Sutherland. General topology, by John Kelly. Topology, by James Dugundji The standard first year undergraduate course in topology continues in the second semester (Math 446) with the development of the fundamental group and covering spaces (aka "Galois theory for spaces"), and then typically gives as an aplication the topological classification of closed surfaces. The textbooks for these courses usually try to straddle the fence between being an introduction to point-set topology, and "algebraic topolgy". Here are two well-regarded books of this type. Topology, (Second Edition) by James Munkres. Topology, by John Hocking and Gail Young Georg Cantor, by Joseph W. Dauben The following web articles, and the references therein, are a good start for further reading. The new links to Banach & Hilbert (and their spaces) are fascinating reading.
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November 20, 2009 - Return to home