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.
|Meeting Time:||MWF 9 - 9:50
|Location: ||Jones 306|
Office: Jones 100D
Office Hours: Mon 2:30-4:30 and Thurs 3:00-5:00.
|Textbook:||Topology, Second Edition, by James R. Munkres|
| Grade |
| 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. |
|Attendance & Lecture Policy:||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.
|Prerequisites: || 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. ||
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):
||Due: Fri, Nov 10
||Wed, Dec 13
||9 AM - 12 Noon
- All relevant announcements will be listed here. Check back frequently (don't forget
to refresh your browser) for updates.
- Important Dates and Class Holidays:
- Fri, Sep 8: ADD/DROP DEADLINE
- 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
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
- (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
- (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
- (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
- (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.)
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
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.
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
|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.