Math 490
Introduction to Topology
Course Information
Tuesday and Thursday 11:30am–12:50pm
East Hall 4096
Professor: Jenny Wilson
Email: jchw@umich.edu
Office Hours: Tuesdays 10:30am–11:20am and Fridays 10am–12pm
Office: East Hall 3863
Course Assistant: Alina Wu
Email: zhixuanw -at- umich.edu
Office Hours: Thursdays 2:30pm–3:30pm
Location: East Hall Atrium
Course Material: The course will investigate ideas relating to open sets, compactness, connectedness, and convergence of sequences in metric spaces and abstract topological spaces. See the LSA course listings for more details.
IBL: Our course will use an
Grades will be made available through Canvas.
Previous years’ webpages:
(2018) http://www.math.lsa.umich.edu/~jchw/2018Math490.html
(2019) http://www.math.lsa.umich.edu/~jchw/2019Math490.html
(2021) http://www.math.lsa.umich.edu/~jchw/2021Math490.html
Textbook: This course has no assigned textbook. Students will develop the class material through guided worksheets. Because the objectives of the course are problem-solving and self-led discovery of the material, students are asked not to use outside sources.
| Grading Scheme: | ||
| Homework | 30% | |
| Class Participation | 15% | |
| Quizzes | 10% | |
| Midterm Exam | 15% | (Tues 22 Oct 2024, in class) |
| Final Exam | 30% | (Fri 13 Dec 2024, 4pm–6pm) |
Attendance policy: Attendance in class is required. From the date a student registers for the class, s/he may miss two classes, but each subsequent unexcused absence will result in a penalty of 3% of the final grade. If you must miss a class, please inform your instructor as early as possible. Circumstances such as illness, job interviews, religious observances, certain university-sponsored events, etc, usually constitute "excused" absences and do not typically count toward the two missed classes, provided your instructor is informed in advance when possible.
Office hours: Attendance in office hours is encouraged! Office hours take place in-person, in Jenny's office (East Hall 3863).
Class conduct: Class discussions and small group work are major components of this course. Students are expected to be active participants in the classroom, and are expected to conduct themselves with professionalism and respect for their classmates. Our goal is to create a supportive class environment where students are comfortable testing ideas, and both offering and receiving constructive criticism from peers.
Homework policy: Each week, students will be asked to write up and submit one or more problems from the latest in-class worksheet(s), along with some additional homework problems. Homework assignments will be posted to the course webpage. Homework is due Fridays at 8pm, and collected through Gradescope. See the Gradescope instructions below. Your homework solutions should be neat and legible.
Students may not consult books or online references (including AI) for their homework. You may work in groups and discuss homework problems with other students, but all solutions (worksheet and homework) must be written up independently and in your own words. You must put away any notes from discussions with classmates or office hours while you write up your solutions, to ensure you fully understand and can reproduce the arguments.
Each student's two lowest homework scores will be dropped.
Quizzes: There will be regular short quizzes throughout the quarter, usually on Tuesdays. Quizzes will typically be between 10-15 minutes. Students will receive advance notice about each quiz and hints about what it will cover. The quizzes are intended to encourage the class to regularly review the material, to provide practice for the exams, and to give feedback (both to you the student, and to me the teacher) about your progress, early on and in a lower stakes setting than the exams.
Each student's two lowest quiz scores will be dropped.
Academic integrity: Students are expected to know and to uphold the LSA Community Standards of Academic Integrity.
Students with documented disabilities: If you might need an academic accommodation based on the impact of a disability, please get in touch with Jenny as soon as possible. Requests for accommodations by persons with disabilities may be made by contacting the Services for Students with Disabilities (SSD) Office located at G664 Haven Hall. The SSD phone number is 734-763-3000 and their website is ssd.umich.edu. Once your eligibility for an accommodation has been determined, this information will be reflected in SSD's Accommodate system. Please note that under most circumstances University Policy is two weeks’ prior notice for any academic accommodation.
Homework
| Homework 0 | Due: Friday 30 August 2024 at 5pm | |
| Homework 1 | Due: Friday 6 September 2024 at 8pm | |
| Homework 2 | Due: Friday 13 September 2024 at 8pm | |
| Homework 3 | Due: Friday 20 September 2024 at 8pm | |
| Homework 4 | Due: Friday 27 September 2024 at 8pm | |
| Homework 5 | Due: Friday 4 October 2024 at 8pm | |
| Homework 6 | Due: Friday 11 October 2024 at 8pm | |
| Homework 7 | Due: Friday 18 October 2024 at 8pm | |
| Homework 8 | Due: Friday 25 October 2024 at 8pm | |
| Homework 9 | Due: Friday 1 November 2024 at 8pm | |
| Homework 10 | Due: Friday 8 November 2024 at 8pm | |
| Homework 11 | Due: Friday 15 November 2024 at 8pm | |
| Homework 12 | Due: Friday 22 November 2024 at 8pm |
| Homework 13 | Due: Friday 6 December 2024 at 8pm | |
| Homework 14 | Due: Tuesday 10 December 2024 at midnight |
Worksheets
| Worksheet 1 | The definition of a metric space | |
| Worksheet 2 | Open subsets of a metric space | |
| Worksheet 3 | Continuous functions of metric spaces | |
| Worksheet 4 | Convergence of sequences in metric spaces | |
| Worksheet 5 | Interior and closure of subsets of metric spaces | |
| Worksheet 6 | Sequential compactness in metric spaces | |
| Worksheet 7 | Complete metric spaces | |
| Worksheet 8 | Homeomorphisms of metric spaces | |
| Worksheet 9 | Abstract topological spaces | |
| Worksheet 10 | The subspace topology | |
| Worksheet 11 | Topological bases | |
| Worksheet 12 | Continuous functions of topological spaces | |
| Worksheet 13 | Convergence of sequences in topological spaces | |
| Worksheet 14 | The product topology | |
| Worksheet 15 | Connected topological spaces | |
| Worksheet 16 | Path-connected topological spaces | |
| Worksheet 17 | Compact topological spaces | |
| Worksheet 18 | Homeomorphisms of topological spaces |
Quizzes
| Quiz 1 | Tuesday 3 September 2024 | (Solutions) |
| Quiz 2 | Tuesday 10 September 2024 | (Solutions) |
| Quiz 3 | Tuesday 17 September 2024 | (Solutions) |
| Quiz 4 | Tuesday 24 September 2024 | (Solutions) |
| Quiz 5 | Tuesday 1 October 2024 | (Solutions) |
| Quiz 6 | Tuesday 8 October 2024 | (Solutions) |
| Quiz 7 | Tuesday 29 October 2024 | (Solutions) |
| Quiz 8 | Tuesday 12 November 2024 | (Solutions) |
| Quiz 9 | Tuesday 19 November 2024 | (Solutions) |
| Quiz 10 | Tuesday 26 November 2024 | (Solutions) |
Exams
The course will have a closed-book midterm exam and a closed-book final exam.
The midterm will be held in-class on Tuesday 22 October, and cover material up to and including Worksheet #7 and Homework #7.
The exam is closed-book, but each student may bring in a single double-sided standard-size (8.5"x11") sheet of notes. Each student must handwrite or typeset their own sheet of notes. Typewritten notes have a minimum of 12pt font.
Some practice material is available:
| 2018 Practice Midterm | (Solutions) | |
| 2018 Midterm | (Solutions) | |
| 2019 Midterm | (Solutions) | |
| 2021 Midterm | (Solutions) |
This year's midterm and solutions are now available.
| 2024 Midterm | (Solutions) |
Our final exam will be held Friday 13 December 4pm to 6pm.
It is a comprehensive exam, covering material from the entire semester, but with more emphasis on material since the midterm.
The final exam is closed-book, but each student may bring in a single double-sided standard-size (8.5"x11") sheet of notes. Each student must handwrite or typeset their own sheet of notes. Typewritten notes have a minimum of 12pt font.
In addition to the Review Package, some previous years' finals are available:
| 2018 Final Exam | (Solutions) | |
| 2019 Final Exam | (Solutions) | |
| 2021 Final Exam | (Solutions) |
This year's final exam and solutions are now available.
| 2024 Final | (Solutions) |
Gradescope Instructions
Gradescope is an online platform for grading homework and exams. Your work is still being graded by a human on the Math 490 instructional team, but Gradescope streamlines the process. Gradescope is designed around grading best-practices, for example, the solutions are anonymized for the grader, points are assigned according to a rubric that we set, and Gradescope allows the grader to give more detailed feedback more efficiently.
You can find instructions and trouble-shooting advice at the Gradescope student centre and Gradescope help page.
How to set up a Gradescope account. Gradescope synchronizes with Canvas to create our course roster. Students should receive an email from Gradescope with information on how to create their log-in credentials. If you have not received this email by Thursday 29 August, contact Jenny.
How to upload an assignment or test. To upload an assignment, you must:
Note: This final step is important! The grader will not see your solution to a question if it is not properly selected.
This video explains the homework submission process.
Resubmission. If you find a mistake in your solution, it is possible to resubmit it anytime before the deadline passes. In Gradescope, click on your assignment, and you will see a "resubmit" button in the bottom right corner. Unfortunately, to make any changes to your solution you must re-upload your whole solution and repeat the page selection process.
How to produce a pdf of your homework. If you write your homework solutions by hand, you can "scan" your solutions to create a pdf. Gradescope has recommendations for apps you can use to produce a pdf with a smartphone. To create legible scans, it is best to write with a dark pencil or pen.
Please preview your scan before you upload it to ensure it is clearly readable.
You can also complete your homework on a computer, using software such as LaTeX. If you do use LaTeX, it may be easiest to draw figures separately by hand and scan or photograph them. You can add graphics to a LaTex document by using the graphicx package, or use software to collate the pdf files after your LaTeX document is complete.
Viewing your graded assignment. Once your work is graded and the grades are "published", you will be able to log into Gradescope to see your graded solutions. Click on the name of your assignment to see a problem-by-problem breakdown of your score. Click on an individual problem to see your solution, the complete grading rubric, and any comments from the grader.
Optional Reading
The following reading is strictly optional: it is not related to the course material and will not be discussed in the course. These are articles on math education and learning psychology which may be of interest to math students.
Dweck - Beliefs about intelligence (Nature.com)
Kimball and Smith - The myth of 'I'm bad at math' (The Atlantic)
Tough - Who gets to graduate (New York Times Magazine)
Paul - How to be a better test-taker (New York Times)
Boaler - Timed tests and the development of math anxiety (Education Week)
Parker - Learn math without fear (Stanford Report)
Steele - Thin ice: stereotype threat and black college students (The Atlantic)
Vedantam - How stereotypes can drive women to quit science (NPR)
Stroessner and Good - Stereotype threat: an overview (University of Arizona)
Lockhart - A mathematician's lament (Mathematical Association of America)
Campus Resources for Wellbeing
As a student, you may experience personal challenges that impacts your ability to participate or impacts your academic performance in our class. These could include anxiety, depression, interpersonal or sexual violence, difficulty eating or sleeping, loss, and/or alcohol or drug problems. The University of Michigan provides a number of resources available to all enrolled students.- UHS After Hours Care information
- Counseling and Psychological Services (CAPS)
- University Health Service 24-hour Nurse Line (734-764-8320)
- Psychiatric Emergency Service (PES) (734-936-5900)
- Sexual Assault Prevention and Awareness Center (SAPAC)
(24-hour Crisis Line: 734-936-3333) - University Psychology Clinic (evaluation and therapy)
- Services for Students with Disabilities (SSD)
- Campus Mind Works (mental health education and drop-in wellness groups
for students with a diagnosed mental health disorder) - Resources for addiction
- Addiction Center and Collegiate Recovery Program
- Wolverine Support Networks
(peer-facilitated group meetings and community events) - UHS Wellness Coaching
- MiTalk
- Helping a friend
- Resources for financial needs and medical assistance
- National Crisis Text Line
- National Suicide Prevention Lifeline (1-800-273-8255)
- Substance Abuse and Mental Health Services Administration Helpline (1-800-662-4357)
- Trevor Project Lifeline (24-hour chat or phone
1-866-488-7386),
and the Trans Lifeline (877-565-8860)