## Math 5853 - 001     Topology I         Fall 2010

### Course Handouts and Messages

• The textbook for this course is Topology (2nd ed.) by James R Munkres. We will work through as much of Part I (General Topology) as possible this semester. Topology II in Spring 2011 will cover topics from Part II (Algebraic Topology). We will also work from chapters 0 and 1 of Algebraic Topology by Allen Hatcher in the Spring 2011 semester.
• Summer 2010 study plan. Read through as much as chapters 1 and 2 as you can. It is important to make sure you know about the typcial material from a discrete mathematics (or foundations) course: sets, subsets, power sets, cartesian products, functions, preimages and images of subsets under functions, order relations, equivalence relations, equivalence classes, partitions, finite and infinite sets, countable versus uncountable sets. Reals and integers, well ordering and induction, principle of recursive definition. You can find a detailed treatment of these topics in Chapter 1 of the text book.
• Summer 2010 study plan. You should also review the notion of a continuous function (as defined in the Calculus sequence or in a foundations or analysis class). If you have seen the notion of a metric space, you could review that too. Don't worry if you have not met this concept. We will treat it in detail in the Fall semester.
• Information Sheet. First week handout.
• Here are some sample questions for mid I.
• Solutions to the midterm exam.
• The (p,q)-torus knot homework can be simplified using the following two useful lemmas. The second lemma uses the following interesing fact about quotient spaces. The scanning process seems to have darkened the following figure in the latter handout.
• Here are copies of my old topology qualifying examinations from Spring 2003 and Summer 2003. You can look here for other qualifying examinations.
• Here is a final examination from Fall 2002.
• You only have to turn in (1) through (9) of the (p,q)-project. You should do so by Monday afternoon (3pm). I will post solutions here once I have all the scripts.

### Homework

• [01]. Due Friday 08/27.   Turn in starred problems.
Pages 14-15:   5*, 7*, 8, 9
Pages 20-21:   1*, 2
Pages 28-29:   2*, 4, 6, 7, 12*.
• [02]. Due Friday 09/03.   Turn in starred problems.
Page 39:   4*, 5
Page 44:   3*, 5, 6, 7
Pages 51-52:   3, 4, 5*, 6*, 7
Page 56:   8
• [03]. Due Friday 09/17.   Write out the details of Proof 8 and Proof 25 from the "Well ordered sets, ordinals" handout.
Write out the details of the circle of implications in Proof 8 of the "Axiom of choice, etc" handout.
Do exercise 11 of the "Axiom of choice,..." handout. I'll give the Axiom of choice handout to you in class on Monday.
• [04]. Due Monday 10/04.   Page 83:   6,7
Page 92:   2, 4, 5, 10
Page 101:   6, 7, 9, 10, 11, 12, 13, 15, 21
Do this problem. There is a small typo in the pdf, $n \in Z -\{0\}$ should read $n \in Z^+ -\{0\}$.
• [05]. Due Monday 10/18.   Pages 144-45:   2, 3*, 4, 6*
Pages 111-12:   2*, 4, 8*, 12, 13*
Page 118:   1, 2, 3, 4*, 5, 6*, 7*
• [06]. Due Monday 11/08.   Page 152:   1*, 5*, 7, 8, 10, 11*
Pages 158-59:   4*, 5, 8, 9*, 10*, 12
Pages 162-63:   3, 4, 5*, 8*
• [07]. Due Wednesday 11/17:   Pages 170-72:   3*, 4*, 5*, 6, 7, 8
Pages 177-78:   1, 3*, 4, 6
Pages 181-82:   6*, 7
• [08]. Due Monday 11/22: Do the following exercises from the Cones, Suspensions, Joins handout:
6, 9, 11, 13, 14, 16, 17, 18 (or 19, your choice).
• [09]. Starred are due by last day of class:   Pages 194-95:   2, 4, 5*, 6*, 7, 11*, 12, 14
Pages 199-200:   1*, 3, 6*, 7
Pages 205-07:   1*, 2*, 3*, 4, 7, 8, 9
• Project.   Turn in the (p,q)-torus knots project by the last day of class.