# Math 426 - Topology - Fall 2015

## General Information:

Meeting Time: MWF 10 - 10:50 Jones 113 Ryan Vinroot Office: Jones 130 Office Hours: MW 11-12, Th 10-11 and 1-4:30 (tentative, also by appt). Topology, Second Edition, by James R. Munkres Class Participation - 5%, Midterm - 30%, Homework - 35%, Final Exam - 30%. The grading scale will be the standard 10 percentage point scale, so that a final score of 93 or higher is an A, 90-92 is an A-, 87-89 is a B+, 83-86 is a B, 80-82 is a B-, 77-79 is a C+, 73-76 is a C, 70-72 is a C-, etc. It is expected that you attend all lectures, with exceptions minimized. It is greatly appreciated when you are on time. Please do your best to stay awake and attentive during lecture, please do not email or text during lecture, and keep all cell phones/hand held devices/laptops put away during lecture. While it is understandable that you may miss a lecture here and there, or be sleepy in class once in awhile, repeated absences, late arrivals, naps, or general non-attentiveness will negatively affect your class participation score. Math 311 Elementary Analysis. It is extremely important to have a thorough knowledge of the topics from Math 214 Foundations of Mathematics. While Math 307 Abstract Algebra is not a prerequisite, we may need the notion of a group for the very last part of the course. It is fully expected that you can write clear proofs without issues, and for this reason another proof-based course like Math 307 is helpful to have prior to this class.
Course Summary: Topology is a tool used to study local information of a space (set with some specified structure). You have seen a very important example of topology in Math 311 Elementary Analysis, namely, the metric topology. Local information of the real line is studied by considering neighborhoods of points. A large part of this class will be the study of General or Point-set Topology, and we will generalize many of the notions and results obtained in Math 311 to a larger class of spaces. Specifically, we will cover the large majority of Sections 12-33 (and Sec. 36, 37) of the text, where Chapter 2 (Sec. 12-22) and Chapter 3 (Sec. 23-29) will develop notions such as continuity, connectedness, and compactness, for arbitrary topological spaces. We will conclude our study of general topology by proving two relatively deep results: The Urysohn Lemma (Sec. 33), which we will apply to introduce imbeddings of manifolds (Sec. 36), and The Tychonoff Theorem (Sec. 37), which is an important result on products of compact spaces. After concluding the above topics on general topology, we will hopefully have a little time to dedicate to an introduction to Algebraic Topology. The main idea of algebraic topology is to construct an algebraic object (such as a group) based on the structure of a topological space, which may be used to compare two topological spaces. We will get as far as we can in Chapter 9 of the book, which gives the construction of The Fundamental Group of a topological space. The only notion needed from Math 307 Abstract Algebra for this part of the course is the definition of a group.

## Dates & Course Announcements:

Midterm and Final Exams:

There will be one midterm, which may have both a timed and take-home component (details will be determined later). The final exam will be timed. The midterm and the final will each count as 30% of your final grade. The final exam will be on Wed, Dec 9, from 2 PM until 5 PM.

Exam Calendar (Tentative):

 Exam Date Time/Due Location Midterm Fri, Oct 23 Mon, Nov 2 Take home Final Exam Wed, Dec 9 2 PM - 5 PM JONES 306
• All relevant announcements will be listed here. Check back frequently (don't forget to refresh your browser) for updates.
• Important Dates and Class Holidays:
• Sat, Oct 10 - Tues, Oct 13: NO CLASS (Fall Break)
• Fri, Oct 23: WITHDRAW DEADLINE
• Wed, Nov 25 - Sun, Nov 29: NO CLASS (Thanksgiving Break)
• Mon, Dec 9, 2:00 PM - 5:00 PM - FINAL EXAM
• (8/26) I will determine my weekly scheduled office hours after the first week or two of classes. During this first (short) week of classes, I will be available in my office on Wed Aug 26, 1-3 pm, and on Thurs Aug 27, 10:30-12 and 1:30-3.
• (8/31) My office hours this week will be: Mon 1:30-3, Wed 1:30-3, and Thurs 10:30-12 and 1:30-3.
• (9/7) My office hours this week will be: Mon & Wed 11-12 and 1-2, and Thurs 10-11:30 and 1-3. I am getting close to figuring out regular office hours (close to these, except eventually split with my other course), so please talk to me if there is an issue with you making it to office hours.
• (9/14) My office hours this week will be the same as last week: Mon & Wed 11-12 and 1-2, and Thurs 10-11:30 and 1-3. Starting next week, I will have several fewer office hours, as my second course will begin.
• (9/21) I've nailed down some likely permanent weekly office hours: MW 11-12 and Th 10-11 and 1-4:30. Please keep in mind that you'll now be sharing office hours with my other course (Math 103 Precalculus).
• (10/14) My Thursday morning office hours this week will be shifted to 9:30-10:30 AM (instead of 10-11). My afternoon office hours will remain the same.
• (11/23) I have listed several optional problems that can be turned in by Wed Dec 2. You can turn in any parts of any of the problems, and it can only count positively towards your total HW grade.
• (11/30) Just to clarify: The optional/extra problems are due at the beginning of class on Wed Dec 2, like normal HW, except I will not give credit for late problems for this one since they are optional.
• (12/3) Our final exam is on Wed Dec 9, 2-5 PM, but not in our lecture room. Our final exam is in JONES 306.
• (12/3) My office hours before the final exam are as follows (apart from my usual Thurs office hours):
Mon, Dec 7: 10-2 and 3-5
Tues, Dec 8: 10-2 and 3-5:30.

## Homework:

Homework problems will be a very important part of the course, and there will be homework assigned almost every week (other than the week of the midterm). Completion of all homework problems is required, and your grade on a homework assignment will be based on completeness, as well as on the details of the solutions of the problems graded. Proofs should be written carefully and neatly, with attention paid to the completeness of your argument. Individual homework assignments should be completed by the student alone, although I am always open for questions, either in office hours or by email. You should not, under any circumstances, attempt to look up solutions or hints to problems online. I will consider this plagiarism, an honor offense.

Homework is due at the beginning of class on the due date of the assignment. Late homework will be marked off 20% for every day late (10% off if it is turned in after the beginning of class, but it is in my hands on the day it is due). Homework turned in after class on the due date is considered one day late, and the next weekday after that 2 days late, and so on. Everything is easier, of course, if you turn in the homework on time!

Homework scores will each be out of 50 points. Your lowest homework score of the semester will be dropped.

 Assignment Problems Due Date 1 pg. 128 #9, pg. 83 #1, 3, 4, 6 Fri, Sep 4 2 pg. 126 #2, pgs. 83-84 #5, 8 pg. 92 #4, 6 Fri, Sep 11 3 pgs. 91-92 #1, 2, pgs. 100-101 #3, 6, 9 Fri, Sep 18 4 pg. 101 #11, 12, 13, pgs. 111-112 #3, 10 Fri, Sep 25 5 pgs. 111-112 #4, 8, pgs. 133-136 #2, 6, 12 Fri, Oct 2 6 pg. 118 #1, 3, pg. 128 #10, pg. 152 #1, 4 Fri, Oct 9 7 pg. 152 #9, 10 Fri, Oct 16 8 pgs. 157-158 #1, 3, pg. 171 #3, 5, 6 Fri, Oct 23 9 pg. 194 #4, 5, 6, 10, 14 Mon, Nov 9 10 pg. 199 #1, 2, 3, pg. 205 #3, 4 Mon, Nov 16 11 pg. 67 #8, pg. 205 #1, 6, pg. 212 #1, 2 Mon, Nov 23 Optional pg. 206 #8, pg. 214 #11, pg. 236 #5 Wed, Dec 2

## Math Major Writing Requirement (Math 300):

If you are a math major, and you would like to complete your major writing requirement through a writing assignment in this class, please let me know in the first week of class. This writing assignment will not count towards your grade in this class, but will rather just serve as your Major Writing Requirement (Math 300). If you decide to do this, you must write your paper on a topic in Topology (or maybe Analysis) approved by me, and you must keep to a schedule of turning in drafts that is set at the beginning of the semester in order to get credit.