# Math 426 - Topology - Fall 2017

## General Information:

Meeting Time: MWF 9 - 9:50 Jones 306 Ryan Vinroot Office: Jones 100D Office Hours: Mon 2:30-4:30 and Thurs 3:00-5:00. 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 is the crucial prereqisite. It is also 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, 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 (that is, a 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. Most of this course will be dedicated 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 midterm will be some time after Fall Break, at the end of October or beginning of November. 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 13, from 9 AM until 12 Noon.

Exam Calendar (Tentative):

 Exam Date Time/Due Location Midterm Due: Fri, Nov 10 9:00 AM Take home Final Exam Wed, Dec 13 9 AM - 12 Noon 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 14 - Tues, Oct 17: NO CLASS (Fall Break)
• Fri, Oct 27: WITHDRAW DEADLINE
• Wed, Nov 22 - Sun, Nov 26: NO CLASS (Thanksgiving Break)
• Wed, Dec 13, 9:00 AM - 12:00 Noon - FINAL EXAM
• (8/28) I will determine my regular weekly office hours soon. My office hours for the first short week of classes are as follows:
Wed Aug 30 2-4 pm, Thurs Aug 31 1-2 pm, Fri Sept 1 2-3 pm.
• (9/4) My office hours this week are as follows:
Mon Sept 4 3:30-4:30 pm, Wed Sept 6 1:30-3 pm, Thurs Sept 7 3-5 pm.
• (9/13) My office hours for the semester are as follows: Mon 2:30-4:30 and Thurs 3:00-5:00. As always, you can email me to make an appointment and we can find some other time that works, just give me a day or so advance notice.
• (9/15) I will be out of town on Mon, Sept 18, and so I will not have my normal office hours that day. Prof. Bolotnikov will sub for me during class that day.
• (10/9) Correction: For HW #6, I mistakenly listed pg. 118 #6 as a problem to do. It should be pg. 118 #7 instead. It has been corrected in the HW list below.
• (10/30) Your take-home midterm is being handed out today. It is due on Fri, Nov 10, at 9 AM in class. The late policy for homework does not apply to the Midterm. The midterm must be handed in on time for credit.
• (12/7) Our final exam has now been officially confirmed to take place in our regular lecture room, Jones 306, and is on Wed, Dec 13, 9 AM-12 Noon.
• (12/8) My office hours during exams prior to our final exam are as follows: Mon Dec 11, 12-2 and 4-5; Tues Dec 12, 12-2 and 3:30-5.
• (12/8) Material for the final exam will cover through the material on Urysohn's Lemma (so all material through the last HW, and all material except for the last week of class). The following is a list of problems that were not assigned for HW, but would be good problems to look at while you study definitions, examples, old HW, and main theorems. These are only meant to give you a skeleton of material from which to study, and is not meant to be a comprehensive list:
pg. 83 #2, 7; pg. 92 #2, 3; pg. 101 #8, 15; pg. 111 #2, 7; pg. 118 #6, 10; pg. 126 #1, 3(b); pgs. 133-136 #1, 12; pg. 152 #1, 12; pg. 158 #6, 10; pgs. 170-171 #1, 7; pg. 178 #4; pg. 194 #4, 10, 12; pg. 199 #4; pg. 205 #2; pg. 212 #3; pg. 235 #2 (This problem is not actually about Tychonoff, it is about Lindelöf spaces.)
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## 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). 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 solutions should be written by students alone, although discussion of the problems amongst students before writing solutions is fine. Also, 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, and if you like you may email me a pdf of your homework if you LaTex it (which is not required but welcomed). Homework that is turned in or in my email inbox 10 minutes after the beginning of the class is considered late. Everyone will be allowed exactly 1 unpenalized late-by-one-day homework (so once during the semester, a HW can be turned in one weekday late by 5 pm with no penalty). After that, late penalties are:
10% off if it is turned in after the beginning of class, but it is in my hands, or in my email inbox as a pdf by 5 pm on the day it is due.
20% off if it is turned in by 5 pm the next weekday after the due date.
20% more off for each (week)day late, turned in by 5 pm, thereafter.
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 8 2 pgs. 83-84 #5, 8, pg. 92 #4, 6, pg. 128 #10 Fri, Sep 15 3 pg. 91 #1, pgs. 100-101 #1, 3, 6, 7 Fri, Sep 22 4 pg. 101 #9, 10, 11, 12, 13 Fri, Sep 29 5 pgs. 111-112 #3, 4, 8, 10, pgs. 133 #2 Fri, Oct 6 6 pg. 118 #2, 3, 7, pg. 134 #6, 8 Fri, Oct 13 7 pg. 152 #2, 3, 4, 7, 9 Fri, Oct 20 8 pg. 152 #10, pgs. 157-158 #1, 2, 3, 9 Fri, Oct 27 9 pg. 171 #3, 5, 6, pg. 177 #1, pg. 186 #5. Mon, Nov 20 10 pg. 194 #5, 11, 14, pg. 199 #1, 2 Fri, Dec 1 11 pg. 205 #1, 3, 4, pg. 212 #1, 2 Fri, Dec 8

## 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.
PLEASE NOTE: If you are a junior, you should not do Math 300 this year, as the new COLL 400 course in the math department which will be offered next year will count as your writing requirement.