# Math 521

## Office hours

M-Th 12:45-1PM in the class room

T and Th 2:15-5Pm in 718 Van Vleck Hall with Chandan Biswas

## Exams

There will be an in-class final on Thursday, 8/10.

## Homework

Please note: It is okay, even encouraged, if you work on the homeworks in groups. The written solutions, though, must be your own. Note that it is painfully obvious when a solution is simply copied and not understood.

There will be regular homeworks posted here:
- Due 6/26: In Edition 1 of Pugh: Chapter 1: 1,2,3,9,10,12,13,14,15,16,*30,36

In Edition 2 of Pugh: Chapter 1: 1,2,3,9,11,13,14,15,16,18,*31,39
- Due 7/3: In Edition 1 of Pugh: Chapter 2: 1, 2, 3, 5, 11, 12, 17, 18, 19, 20, 22, 23, 29, 30, 31

In Edition 2 of Pugh: Chapter 2: 23, 24, 25, 26, 29, 40 (a,b,c,e), 13, 14, 15, 16, 18, 19, 39, 9, 10, 11(a,b)
- Due 7/10: In Edition 1 of Pugh: Chapter 2: 34,(a-e) , 14, 25

In Edition 2 of Pugh: Chapter 2: 28(a-e), 30, 31, 34,55
- Due 7/17: If every closed and bounded subset of a metric space M is compact, does it follow that M is complete? (Proof or counterexample.) , Show that the 3 metrics we discussed on the product space $X\times Y$ are in fact metrics, 38, 39, 40, 27

In Edition 2 of Pugh: Chapter 2: 22, 38, 41, 43, 44, 52
- Due 7/26: In Edition 1 of Pugh: 6, 10, Prove that if $A$ and $B$ are compact, disjoint, non-empty subsets of $M$, then there are points $a\in A$ and $b\in B$ so that for any $x\in A$ and $y\in B$, $d(a,b)\leq d(x,y)$. In other words, $a$ and $b$ are the closest you get a point in $A$ and a point in $B$ to each other., 28, 43,14,41,54,55,56,59,64a, 68,69,74,77

In Edition 2 of Pugh: 27,33,46,48,53,55,56,57,58,59,62,66a,70,71,76,77
- Due 8/2: In Edition 1 of Pugh: 44-53, 109(a-d), 118,89,91, 92, 130

In Edition 2 of Pugh: 85-94, 99(a-d),103,122, 124, 125, 152
- Due 8/14 (Either to Chandan Biswas's mailbox on the 2nd floor of Van Vleck or scanned and e-mailed to cbiswas@math.wisc.edu):

In Edition 1 of Pugh: 1, 2, 3, 5, 7, 8, 28, 30, 32, 34, 50, 51, 52, 53

In Edition 2 of Pugh: 1, 2, 3, 5, 6, 7, 27, 29, 31, 33, 51, 52, 53, 54,

*Problems with a star are either harder or require some more serious detail-work.
## Instructions from the grader:

Recall that this is a proof-based course, and thus a strict level of rigor is necessary in your solutions. Pictures are never proofs, but are always encouraged along with proofs.

Please put the problems in order and staple your homework in the upper left corner. UNSTAPLED HOMEWORK will be creatively dealt with at the discretion of the grader. Please use complete sentences. Your solution should not be just a sequence of equations and formulas. Make sure that your presentation is clean. Do not submit solutions with crossed out parts. Recopy problems if necessary.