# Math 426 - Topology - Fall 2013

## General Information:

Meeting Time: MWF 9 - 9:50 Jones 113 Ryan Vinroot Office: Jones 130 Office Hours: M 10-11 and 4-5, W 10-11 and 3-4:30, Th 10-12 (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. By the end of the course, we will also need a few very basic notions from Math 307 Abstract Algebra (essentially, if you know what groups and homomorphisms are by November, you will be fine).
Syllabus: Topology is a tool used to study local information of a space. 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 studying 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 essentially all 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 begin 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.

## Dates & Course Announcements:

Midterm and Final Exams:

There will be one midterm, and it will be a take-home midterm. 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 Mon, Dec 16, from 9 AM until noon.

Exam Calendar (Tentative):

 Midterm Handed out Oct 28 Due Nov 4 Take home Final Exam Mon, Dec 16 9 AM-12 noon Jones 113
• 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 12 - Tues, Oct 15: NO CLASS (Fall Break)
• Fri, Oct 25: WITHDRAW DEADLINE
• Wed, Nov 27 - Sun, Dec 1: NO CLASS (Thanksgiving Break)
• Mon, Dec 16, 9:00 - 12:00 - FINAL EXAM
• (8/28) 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 for any questions on Wed, Aug 28, 10-11 and 2:30-4, and Thurs, Aug 29, 9:30-11:30.
• (9/2) Here are some office hours for this week: Mon, Sept 2, 10-11 and 3-4:30; Tues, Sept 3, 9:30-11; Wed, Sept 4, 10-11 and 3-4:30; Thurs, Sept 5, 10:30-12.
• (9/6) My office hours next week are: Mon, Sept 9, 10-11; Tues, Sept 10, 10-11:30, Wed, Sept 11, 10-11 and 3-4:30; Thurs, Sept 12, 10:30-12.
• (9/9) I had to slighlty shift my Tues, Sept 10 office hours to 10-11:30 AM (rather than 9:30-11 AM). I apologize for any inconvenience this may cause.
• (9/13) I have set my weekly office hours for the semester as: Mon 10-11 and 4-5, Wed 10-11 and 3-4:30, Thurs 10-12.
• (10/7) My office hours this Wed afternoon (10/9) have to be shifted earlier, and will be 2:15-3:45 (instead of 3-4:30).
• (10/16) Please note that HW #7 is due on Wed, Oct 23 (instead of the usual Friday). The take-home midterm will be handed out on Mon, Oct 28, and will be due on Mon, Nov 4.
• (11/4) My afternoon office hours today, Nov 4, will be 3-4 instead of 4-5.
• (12/6) I will have your last homework graded and ready for you to pick up in my office on Monday, Dec 9. You can come pick it up any time during the hours 12-2 or 3-5 in my office (Jones 130).
• (12/6) During the first week of finals (prior to our final) I will be available in my office on the following days and times: Mon, Dec 9, 12-2 and 3-5; Thurs, Dec 12, 9-11, 12-2, and 3-5; Fri, Dec 13, 9-11, 12-2, and 3-4:45.
• (12/12) My office hours on Fri, Dec 13, are ending at 4:45 instead of 5.

## 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. 83 #1, 3, 4(a,c), 5, 6 Extra: 4(b) Fri, Sep 6 2 pg. 91-92 #1, 4, 6, 8, 9 Fri, Sep 13 3 pg. 100-101 #1, 3, 6, 11, 13 Extra: pg. 102 #21 (no due date) Fri, Sep 20 4 pg. 111-112 #3, 4, 7, 8, 10 Fri, Sep 27 5 pg. 118 #1, 3, 10, pg. 134 #8 Fri, Oct 4 6 pg. 152 #4, 9, 10, pg. 158 #9, 10 Fri, Oct 11 7 pg. 171-172 #3, 5, 6, 7, 11 Wed, Oct 23 8 pg. 194 #5, 6, 10, 11, 14 Fri, Nov 15 9 pg. 199 #1, 2, pg. 205 #1, 3, 4 Fri, Nov 22 10 pg. 235 #2, pg. 212 #1, 3, pg. 227 #5 *Extra: pg. 214 #11 Fri, Dec 6

## 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. If you decide to do this, you must write your paper on a topic in Topology 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. You are also encouraged to sign up for Math 300 during this semester if you fulfill the writing requirement through this class.