There will be homework assigned every week, due on Thursdays

Homework #1; (due Thursday, January 17):  State a mathematical fact or formula that you find interesting, and prove it.  You can assume all reasonable facts and theorems that lead up to your result, but you should make clear what you are assuming.   This assignment will be graded and everybody who makes a credible effort will get 100%.  I mostly want to see the extent to which you can write a clean proof.   [Warning: the most trivial facts are often the hardest to prove, because there's so little that you can assume.  Don't try proving that 2+2=4, unless you're willing to give precise definitions of "2", "+", "=" and "4"!]

Homework #2; (due January 24)
Page 14, problems 2, 4, 6
Page 20, problems 1, 2, 3, 6

Homework #3: (due January 31)
Page 28, problems 1, 3, 6, 12, 13
Page 34, problems 4, 6, 11

There is no homework due the week of February 7, but here are some PRACTICE problems you way wish to look at.
Page 44, problems 1, 3, 5 (of course it's true, but how do you prove it?)
Page 51, problems 5,6,9
Page 61, problems 1, 2, 7

Homework #4: (due February 14)
Do all of the problems on the midterm over again, and submit CLEAN solutions, preferably based on the ideas you had during the test (if they were on the right track). You can find the exam on the class web page. To avoid temptation, I have temporarily taken the solutions off the web. On problem 3, the handwaving that I said was OK for the test is not OK for homework. Take the time to write things up precisely.

Homework #5: (due February 21)
Page 83, #2, 5, 8
Page 91, #2, 3, 5, 8, 9, 10. On problem 8 there are three cases to consider, depending on whether the line is horizontal, vertical, or neither.

Homework #6: (due February 28)
Page 100, #1, 5, 6, 13, 20
Page 111, #2, 6, 8, 10.

Homework #7: (due March 6)
Page 118, #5, 6, 7, 8
Page 126, #1, 2, 4, 5, 6, 8.

Homework #8: (due March 20) (Can be turned in March 25)
Page 126, #11
Page 133, #3, 7
Extra 3-part problem:
A. Let X=RxR, and let (x,y)~(x+n,y+m) for all integers n,m and real numbers x,y. Describe the quotient space and its topology.
B. Again, let X=RxR, and let z be a nonzero number. Let (x,y)~(x+t,y+zt) for every real number t. Describe the quotient space and its topology.
C. Now combine the last two exercises. Let (x,y)~(x+n+t,y+m+zt). What is the quotient space if z is rational? What is the topology of the quotient space if z is irrational? (The space itself is hard to describe, but the topology is simple)

Homework #9: (Due TUESDAY April 1)
Page 152, #1, 2, 5
Page 157, #1, 3
Page 162, #2
Page 170, #1, 4, 6

There is no homework due the week of April 8, because of the midterm that day, but here are some PRACTICE problems you way wish to look at.
Page 170, #5, 10
Page 177, #2, 6
Page 181, #1, 2, 7

Homework #10: (Due THURSDAY, April 17)
First, write up a clean set of solutions to the midterm. Try to do them on your own, but you can peek at the posted solutions if necessary. Then do all of the posted review problems: Page 170, #5, 10
Page 177, #2, 6
Page 181, #1, 2, 7

Homework #11: (Due THURSDAY, April 24)
Page 194, #1,4, 15
Page 199, #1, 4, 7abd (skip part c)
Page 205, #1, 2

Homework #12: (Due THURSDAY, May 1)
Page 205, #5
Page 212, #1, 2a, 4, 5
Page 218, #1, 3