Welcome to the wonderful world of topology!
Topology, more or less, is the study of spaces, or more specifically spaces where you can reasonably talk
about continuity and convergence. Topological spaces can be quite crazy, but we will figure out how to deal with them and understand when they are not so crazy. Some of the most basic questions we can ask are what do these
things look like? How many are there? How can we tell two of them apart? What interesting properties do they have?
Among other things we will see how to recognize nice spaces from minimal data, prove that "Cantor dust" can block out all light from the sun, show that a 2 dimensional square is the continuous image of a small line segment, and construct nowhere differentiable functions.
The course will cover:
- Topological spaces and Continuous functions
- Examples and constructions of topological spaces
- Topological properties: connectedness, separation axioms, countability properties, compactness
- Metrics spaces and completeness
- Classification results: Cantor sets, surfaces
Announcements:
- Please take part in the Course/Instructor Opinion Survey (CIOS).
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- The final exam will be Friday, December 9th, from 11:30 to 2:20. The room is Skiles 171 (to be confirmed). The exam will be comprehesive covering all the materical in the class. The best way to prepare for the test is to (1) go through the class notes, (2) go through all the homework problems, (3) talk to me if you have questions or read some of the recommended sources. Here is some information about the test:
- The final exam will consist of 6 problems, plus one problem consisting of some true/false questions.
- Most will be similar to the homework problems.
- As a "practice test" work the questions on the previous two test, plus the practice problems for the tests and in addition work problems 6, 12, 13, 14, and 15 on HW #5.
- I guarantee that at least 3 problem on the test will be either be among these problems or very similar to them. These "guaranteed problems" will be evenly distributed, that is at least one will be from the material covered by the first test, one from the second test and one from the material since the second test.
- The second midterm exam will cover the material in Section III and IV in the notes and on the homework assignments 3 and 4. The best way to prepare for the test is to (1) go through the class notes, (2) go through all the homework problems, (3) talk to me if you have questions or read some of the recommended sources. Here is some information about the test:
- The test will be approximately 4 or 5 questions.
- Most will be similar to the homework problems.
- As a "practice test" work problems 1, 2, 6, and 9 from homework 3 and problems 3, 10, 11, and 13 from homework 4.
- I guarantee that at least 2 of the problems on the test will either be among these problems or very similar to them.
- One of the questions on the test will consist of several True/False or short answer questions.
- The second test in the class will be in class on November 14th.
- The first midterm exam will cover the material covered through Theorem 14 in Section III in the notes and on the homework assignments 1 and 2. The best way to prepare for the test is to (1) go through the class notes, (2) go through all the homework problems, (3) talk to me if you have questions or read some of the recommended sources. Here is some information about the test:
- The test will be approximately 4 or 5 questions.
- Most will be similar to the homework problems.
- As a "practice test" work problems 5, 8, and 14 from homework 1 and problems 7,8, and 11 from homework 2. (Note I updated problem 11 on homework 2.)
- I guarantee that at least 2 of the problems on the test will either be among these problems or very similar to them.
- One of the questions on the test will consist of several True/False or short answer questions.
- The first test in the class will be in class on October 3rd.
Course Information:
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