Table of topics and assignments 

 

If you are rusty with MAA 3200, you should review the appendices (A, B, C for now). Appendix C is very important, you should read it carefully and do all exercises, if you are not well familiar with the notions there. I presume you are familiar with the notions in Appendices A and B (if not, you should talk to me during the first week of classes). You should still read them to get used to the notations.

 

Date Topics covered Suggested Assignment Comments
Aug. 25 1.1 Metric Spaces 1.1 all exercises  and

all exercises from Appendix C		 Hint for 19: show that |f(x) - f(y)| is less or equal than d(x,y).

read also the theory of Appendix C
Aug. 27 1.2 Topological spaces
1.3 (part of) Basis		 1.2 all exercises
1.3 # 4, 6, 8, 11, 12, 13 (more below)		 Homework 1 is due Thursday, Sept. 3. (typo corrected now)
Sep. 1 1.3 (rest of) Sub-basis, Countability Axioms
(the material on pages 23, 24
on order topology will not be required)		 1.3 # 1, 2, 3, 5, 9, 15, 18, 22 Also read Appendicies E, F and G
Sep. 3 1.4 (part of) Closed Sets, Closure, Interior




 1.4 #1-7, 9-16, 18-25, 28-30

For grad students and all undergrad students interested
in doing a project, please look over the attached topics
and let me know by next Thursday what topic you choose.
If you want to propose a different topic, that is possible,
but show me a sketch of what you plan to cover.

For undergraduate students still thinking about the project option,
please make your choice known to me and the rest of the team by  Thursday, Sep. 17.

 Project topics assignments (first in the list is the graduate student in the team)
(1) Kuratowski 14 sets theorem (Pbs. 31, 32 section 1.4 textbook
+ online material + possibly Pbs. 26, 27 same section)
   Juan Castro, Ronny Balboa,
(2) The Weak Topology (section 2.4 + some exercises)
K.C. Mangen
(3) Separation axioms (selection of 5.1 and 5.2 + some exercises)
   Giancarlo Sanchez, Sandra Figueroa,  Alberto Mizrahi
(4) Normed spaces (selection of section 6.2 + some exercises)
   Michael Davidson, Hector Sanabria
(5) Manifolds (selection of section 6.4 + some exercises)
    Lazaro Diaz, Johnny Fonseca
(6) Fractals (selection of section 6.5 + some exercises)
    Jorge Rivero, Caitlin Hogan, Antonio Ruiz
(7) A continuous function on [0,1] not differentiable anywhere (Example 55, Theorem 1.51 text
+ associated exercises) Eoin Moore,
Sep. 8 (rest of) 1.4 1.4 #1-7, 9-16, 18-25, 28-30 Solution key for Homework 1
Sep. 10 1.5 More on metric spaces 1.5 #1, 2, 3 (done already in hwk 1), 4-6, 9-14

Read on your own Thm. 1.30 -- it is very similar to a problem in your hwk 1 related to Example 49.		 Homework 2 is due Thursday, Sep. 17 updated version
There was a mistake in 4(c) of the initial version of this homework. The statement is now corrected. I also added a question 4(d) and some clarifications of definitions for Pb. 2.
Please check the updated version of the file.
Sep.15 (part of) 1.6 Convergence 1.6 #1-6, 13(obvious typo in (c)) Read on your own the proof of Theorem 1.46.
The material on pages 49-50 of this section is not required.
It is the subject of the project (7) above.
Sep. 17  1.6 Baire's Theorem
(part of) 1.7 Continuous functions		 1.7 #1-21 Solution key for Homework 2
Sep. 22 (rest of) 1.7 Homeomorphism 1.7 #1-21 I have note yet covered Theorem 1.58, but will do so later.
It is an important result, so you may read it on your own even at this point.
Sep. 24 2.1 Subspace Topology
 2.1 #1-9, 13-18, 21, 22 Homework 3 is due Thursday, Oct. 1
Sep. 29 2.2 Topology on X×Y 2.2 #1-11, 13-16 Solution key for Homework 3
Oct. 1 2.3 Infinite products 2.3 #1-6, 8, 10, 11, 13-15 Additional practice problems for the midterm -- others might be added later
Oct. 6 2.5 The uniform metric 2.5 #1, 2, 5, 6, 8, 9 (do these after the midterm) The midterm on Tuesday, October 13, covers all material up to (and including) section 2.3,
but does not include section 2.5.
Oct. 8 Problem solving   For the exam, you should know well the definitions of all concept discussed.
Most questions will be common sense proofs that should come 
naturally from the definitions. Many of the theorems that we've done (or left as exercises)
fall into this category. Any of the suggested exercises, the additional practice problems above, or the hand-in homework problems could serve as exam questions as well. 
Oct. 13 Midterm Solution key for midterm  
Oct. 15 More on 2.5
 2.5 see problems above
  
Oct. 20 2.6 Quotient spaces 2.6 #1-7, 10-12, 14-17 (you may rephrase #3 and 4 in terms of equivalence relations rather than partitions) Homework 4 revised -- due Tuesday, October 27.
Thanks to Ivan Gonzalez for pointing out the mistake in the last problem (previous version).
That problem is now optional (for bonus points).
Oct. 22 More on 2.6 2.6 see above problems Solution key for Homework 4
Oct. 27 3.1 Connected Spaces 3.1 #1-14, 18-22 New Office Hours: Tuesdays, Thursdays 1:30-3:30pm
(starting Oct. 27 until the end of the semester)
Oct. 29 More on 3.1 see above problems  
Nov. 3 3.2 Pathwise and local connectedness 3.2 #1-5, 7-9, 11 Quiz on Thursday, Nov. 5, from sections 3.1 and 3.2
You'll have to write a couple of definitions and
answer one or two quick questions.
Nov. 5  Compactness 1

Quiz 1		  #1,2, 3, pages 170-171 in the Munkres handout

Solution key for quiz 1		 Munkres sections on compactness

For compactness, I will generally follow the presentation in Munkres,
(see the link to the pdf above), but with some references to results
and exercises from Patty as well.
Nov. 10 Compactness 2 #1-9, pages 170-171 Munkres
#9, 13, page 147 Patty		  
Nov. 12 Compactness 3 #1-7, pages 181-182 Munkres  
Nov. 17 Compactness 4 #2-6, pages 177-178 Munkres
(pb. 3 added)		 Quiz on Thursday, Nov. 19, from compactness, sections 26, 27, 28, Munkres.
You'll have to write a couple of definitions and answer one or two quick questions.
Nov. 19 Quiz 2    
Nov. 24 5.5 Urysohn's Lemma,
Tietze's Extension Thm		 5.5 (Patty) #1-5 Projects are due on Dec. 1 (no delay please).
Nov. 26 No Class.
Happy Thanksgiving!		    
Dec. 1 Projects due
Project presentations		    
Dec. 3 Review for final exam Note: I add Pb. 3, page 177 Munkres to your suggested exercises. The final exam will have a certain emphasis on the material not covered on the midterm
(that is starting with section 2.5 on). However, the final is cumulative and there will be questions related to the earlier material. In particular, you should know well the definitions of all concepts covered.
Most questions will be common sense proofs that should come naturally from the definitions.
Many of the theorems that we've done (or left as exercises) fall into this category.
Any of the suggested exercises, the additional practice problems above,
or the hand-in homework problems could serve as exam questions as well. 
For everyone doing a project, on the final there will be one additional problem, project specific.
The score on this question will be part of the project grade, but will not affect the score on the final exam.
Dec. 8 Final Exam
9:15-11:45
regular room		    
    Happy Holidays! These are your scores on the final exam out of 110 points (second column)
and your grade for the class (last column).