Past Courses
Urbana, Illinois
Fall 2009
Topology - Math 460
Textbook:
Introduction to Topology: Pure and Applied, by Colin Adams and Robert Franzosa.
A list of known errors in the textbook can be found here.
Course Description:
Topology is a relatively new branch of geometry that studies very general properties of geometric objects, how these objects can be modified, and the relations between them. Three key concepts in topology are compactness, connectedness, and continuity, and the mathematics associated with these concepts is the focus of the course. Compactness is a general idea helping us to more fully understand the concept of limit, whether of numbers, functions, or even geometric objects. For example, the fact that a closed interval (or square, or cube, or n-dimensional ball) is compact is required for basic theorems of calculus. Connectedness is a concept generalizing the intuitive idea that an object is in one piece: the most famous of all the fractals, the Mandelbrot Set, is connected, even though its best computer-graphics representation might make this seem doubtful. Continuous functions are studied in calculus, and the general concept can be thought of as a way by which functions permit us to compare properties of different spaces or as a way of modifying one space so that it has the shape or properties of another. Economics, chemistry, and physics are among the subjects that find topology useful. The course will touch on selected topics that are used in applications. Prerequisite: MATH 341 or permission of instructor.
Syllabus
Homework:
Assignments will be announced in class and then usually posted here. The classroom announcement serves as official notification.
Monday, August 31st
Make sure you are familiar with material in sections 0.3 - 0.6.
Due Friday, September 4th
1.3, 1.5, 1.6, 1.7, 1.9
Due Wednesday, September 9th
1.11, 1.16, 1.17, 1.20-22
Due Wednesday, September 16th
1.27a, 1.35, 1.36, 2.1 (answers only), 2.7, 2.10, 2.12
Due Monday, September 21st
2.19, 2.20, 2.21, 2.23, 2.24 (answers only)
Due Friday, September 25th
3.7, 3.10, 3.11, 3.19, 3.20, 3.22
Due Monday, September 28th
3.25, 3.27, 3.28, 3.33, 3.35, 3.37
Due Wednesday, October 14th
4.6, 4.7, 4.9, 4.10, 4.27, 4.29, 4.30, 4.34
Due Monday, October 19th
5.2, 5.5, 5.6, 5.10, 5.12, 5.15
Due Wednesday, October 28th
5.23, 5.25, 5.29 (part b not true, give counterexample), 5.31
Due Wednesday, November 4th
Problems in class email
Due Friday, November 13th
6.1, 6.7, 6.9, 6.10b
Due Monday, November 30th
6.17a, 6.18, 6.19, 6.30, 6.34, 6.41, 6.44
Due Wednesday, December 9th
7.5, 7.13a, 7.21, 7.25, 7.35
Due Monday, December 14th
7.41, 8.12, 10.4, 14.2
Lecture Topics:
Each day’s lecture topic will be posted here after class. A tentative schedule of what’s to come is in the syllabus.
Mon, Aug 31: Introduction to Topology
Wed, Sep 2: 1.1 Open sets and the Definition of a Topology
Fri, Sep 4: 1.2 Basis for a Topology
Mon, Sep 7: 1.3 Closed Sets
Wed, Sep 9: 2.1 Interior and Closure of Sets
Fri, Sep 11: 2.2 Limit Points
Mon, Sep 14: 2.3 The Boundary of a Set
Wed, Sep 16: 3.1 The Subspace Topology
Fri, Sep 18: 3.2 The Product Topology
Mon, Sep 21: 3.3 The Quotient Topology
Wed, Sep 23: 3.4 More Examples of Quotient Spaces
Fri, Sep 25: 3.4 More Examples of Quotient Spaces
Mon, Sep 28: 4.1 Continuous Functions
Wed, Sep 30: Review, Exam 1 Handed Out
Fri, Oct 2: 4.1 Continuous Functions, Exam 1 Due
Mon, Oct 5: 4.2 Homeomorphisms
Wed, Oct 7: 4.2 Examples of Homeomorphisms
Fri, Oct 9: 4.2 Homeomorphisms and Embeddings
Mon, Oct 12: Reading Days - No Class
Wed, Oct 14: 5.1 Metric Spaces
Fri, Oct 16: 5.1 Metric Spaces
Mon, Oct 19: 5.2 Properties of Metric Spaces
Wed, Oct 21: 5.2 Properties of Metric Spaces
Fri, Oct 23: Countability Axioms
Mon, Oct 26: Countability Axioms
Wed, Oct 28: Countability and Separation Axioms
Fri, Oct 30: No Class (MAA Section Meeting)
Mon, Nov 2: Separation Axioms
Wed, Nov 4: Review, Exam 2 Handed Out
Fri, Nov 6: Urysohn's Lemma and Metrization Theorem, Exam 2 Due
Mon, Nov 9: 6.1 Connectedness
Wed, Nov 11: 6.1-2 Distinguishing Topological Spaces via Connectedness
Fri, Nov 13: 7.1 Open Coverings and Compact Spaces (Zoey)
Mon, Nov 16: 6.3 Intermediate Value Theorem
Wed, Nov 18: 6.4 Path Connectedness
Fri, Nov 20: 7.2 Compactness in Metric Spaces
Mon, Nov 29: 7.3 Extreme Value Theorem and Lebesgue Numbers
Wed, Dec 2: 8.1 Iterating Functions (Chrissy)
Fri, Dec 4: 7.4 Limit Point Compactness
Mon, Dec 7: 10.1 The Brouwer Fixed Point Theorem (Yaowen)
Wed, Dec 9: 7.5 One-Point Compactifications
Fri, Dec 11: 14.1 Manifolds (Ben)
Mon, Dec 21: Final Exam
Chris Camfield, Visiting Assistant Professor