MATH 320/620
Math 320/620, Sec 01, Point-Set Topology, Spring 2024
Prelims:
Assignments:
Will be posted on this website after each class,
generally due the next class.
- Due 4/1. Do problems in 3.4 #7, 8, 12. (see also #1, 4; essentially done
in class). Read 3.6 and do problems # 3, 6. Students in 620 should be able
to prove Theorem 5.5 on their own. (It is a good
time to start reveiwing previous homeowork assignments.)
- Due 3/27. Read 3.4 and do problems 3, and write a formal proof of
Theorem 4.12 and Corollary 4.13. Read 3.5.
- Due 3/25. Review notes on bases, and finer/coarser topologies. (This
subsumes material in 3.3; don't read 3.3.) Do the two
problems given in class. 1) For X={a,b,c} with topologies C1 and C3 given in
class, show the given function is a homeomorphism but the identity function
is not continuous. Show the collection of open arcs on the circle is a
basis. Is the topology that it generates the same as the topology of the
circle induced by the Eulidean metric on the plane restricted to the circle?
- Due 3/20. Review you notes from last class, and be sure to understand the
definition of basis, and how it determines a topology. Do the following
problem: Let X be the extended real numbers. Verify the properties for a
basis, as given in class. Are the real numbers a closed subset? Show that
positive and negative infinity are limit points. Conclude that the only
closed subset of the extended reals that contains the reals is the extended
reals itself.
- Due 3/18. Read 2.6 again as needed. Which of the following are closed
subsets of the real numbers (with its standard topology): the integers, the
rationals, the set of all real numbers which are solutions to some polynomial equations with integer coeficients? For each, try to give two proofs, using hte definition of
closed (i.e. the complement is open) or using limit points.
Show that if f:R \to R is continuous then the graph of f is a closed subset
of the Euclidean plane. The converse is not true: do problem #4.
(The problem from class): Let A be subset of a metric space (X,d). Is it true that if a point b in A is limit of a
sequence in X, then b is a limit point of A?
- Due 3/13. Read 2.6. Do problems in 2.4 #3, 4. Show that (a,b] is neither
open nor closed, as a subset of the real numbers with the standard topology
induced by the Euclidean metric. Let (X,d) be a metric space with d(x,y) = 1
iff x is not equal to y. Determine what are the convergent sequences in X.
- Due 3/11. read 2.5 and do problem 2. Read 2.6 and do problem 2. Read
3.1
(This material in 3.1 will take some time to digest, so I've discussed it once in class,
asked you to read it, and we'll let it sink in while we continue more topics
in Ch 2.)
- Due 3/6. Read 2.4, especially Definitions 4.4 and 4.9. Do problems 1, 4, 5 (hint
consider Bn=B(a,1/n)).
- Due 2/28. Read 2.4 and do problems #2, 6.
- Due 2/26. Read 2.3 and do problems 1, 3. Read 2.4.
- Due 2/21. Read 2.1 and 2.2 and do problems in 2.2 #2, 3, 4,
7, 8. Students in 620, do #5 and 6.
- Due 2/14. Do problems in 1.6 #1 ,2 ,3, 4 to be handed in. You'll have
additional tries, as needed, to complete them. These must be mastered. Here
are solutions.
- Due 2/7. Do problems in 1.4 #4,5, and read 1.8 and do problem 4. Start on
problems in 1.6 #1,2,3,4.
- Due 2/5. Read 1.4 and do problems 1a,b,f (first part), 3b. Read 1.5 and
do problem 1. Read 1.6 and do problem 1a,b. (Do also any previous problems,
as needed.)
- Due 1/31. Read Ch 1.1 to 1.3, and do problems in 1.2 #1, 2, 3. and 1.3, #1.
Course Information:
This course is an introduction to the basic concepts and fundamental results
of point-set topology. The course includes a review of sets and functions, as
well as the study of topological spaces including metric spaces, continuous
functions, connectedness, compactness, and elementary constructions of
topological spaces. Additional topics such as the concept of homotopy, Euler
number, and fundamental group, may be included as time permits.
The expected course outcome is for the student to learn this material and
demonstrate their level of mastery on the course assessments. This should be accomplished by being an active participant in class,
completing all of the assignments, studying independently and in groups
outside of class, and seeking
my help as needed.
- Location: Mondays and Wednesdays, 10:45-12, Kiely Hall Rm 422. In person.
- Textbook: "Introduction to Topology" 3rd Edition, by Bert Mendelson. As a
second reference (not required) is "Introduction to Topology"
2nd Edition, by Gamelin and Greene.
- Grading and Assessments: The final grade will be based on two preliminary exams (30%
each), and the final exam (40%).
- Exams: The two preliminary exams are in class on 3/4 and 4/10.
The final exam is Monday, May 20th, 11am-1pm in Kiely Room 422.
Contact Information:
- Instructor: Prof. Scott Wilson
- Email: scott dot wilson AT qc.cuny.edu
- Office hours: Mon 12:45-1:30 (Kiely 609) and Wed 12:45-1:30 (Kiely 331- Mathlab), or by appointment.
- Office: Kiely 609
Note on cross-listed courses:
This course has been cross-listed as both an undergraduate course MATH 320 and
a graduate course MATH 620. The class sessions will be the same for students
enrolled in the two classes; however, students enrolled in MATH 620 will have
higher expectations, as indicated in homework assignments and exams. Graduate
students must enroll in MATH 620 and not MATH 320. Undergraduate students may
decide to enroll in either MATH 320 or MATH 620. If an undergraduate student
enrolls in MATH 320, they will not be able to enroll in MATH 620 in the
future, and MATH 320 cannot count toward a graduate degree at Queens
College. If an undergraduate student enrolls in MATH 620 and is an Accelerated
Masters student, MATH 620 may count as 3 of the maximum 12 credits toward
their Masters Degree. If an undergraduate student enrolls in MATH 620 and is
NOT an Accelerated Masters student AND MATH 620 is not one of the courses that
fulfills their MATH degree requirements AND if MATH 620 is not used toward the
120 credits in their undergraduate degree, then MATH 620 may count as 3 of the
maximum 12 credits toward an eventual Queens College Masters Degree. Please
contact Prof. Wilson if you have any questions about this policy.
Academic Integrity:
In case of cheating or plagarism, I will seek
academic and disciplinary sanctions. See The CUNY Policy on Academic Integity.
Reasonable Accommodations for Students with Disabilities:
Candidates with disabilities needing academic accommodation should: 1)
register with and provide documentation to the Office of Special Services,
Kiely Hall Room 108; 2) bring a letter indicating the need for accommodation
and what type. This should be done during the first week of class. For more
information about services available to Queens College candidates, visit Office of Special Services for Students with Disabilities in Kiely Hall Room 108 or contact the Director, Dr. Miriam Detres-Hickey at QC.SPSV@qc.cuny.edu.
Statement on Student Wellness:
As a student, you may experience a range of challenges that can interfere with
learning, such as strained relationships, increased anxiety, substance use,
feeling down, difficulty concentrating and/or lack of motivation. These mental
health concerns or stressful events may diminish your academic performance
and/or reduce your ability to participate in daily activities. QC services are
available free of charge. You can learn more about confidential mental health
services available on campus at Counseling Services Department.
Course website: https://qcpages.qc.cuny.edu/~swilson/math320.html