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Mathematics Department - University of Michigan




Topology is a fundamental area of mathematics that provides a foundation for analysis and geometry. Once a set has a topology (so it becomes a "topological space"), we can start to build on it. For example, the notion of a continuous function makes sense on a topological space, and in fact, this is the most general setting where the idea of a continuous function makes sense. One theme in topology is trying to distinguish topological spaces from each other. A topologist will tell you that all pentagons look the same, and in fact, that all pentagons look like all triangles! One thing we will do in this course is rigorously explore what it means for two topological spaces to be "the same". We will also develop tools that will help to distinguish topological spaces from each other. Topics include: open/closed sets, metric spaces, continuity, homeomorphisms, connectedness, compactness, Euler characteristic. This course is taught in the IBL style. This means that the instructor will speak only for a few minutes at the beginning of class, and then the students will work in groups (guided by worksheets) to explore and develop the material. There is no textbook; the main reference for course material is the compilation of the worksheets from class. By group discussions and problem-solving, students will discover the world of topology.

The prerequisites for Math 490 are: Math 351, Math 451 or previous exposure to real analysis. Even if you have not taken these courses, it is still possible to take 490 with permission of the instructor. Lastly, group work is a large part of this course, so it is important that you can work effectively with your peers.

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