Math 190A (Winter 2021)
Summary
Math 190A is the first course in the Introduction to Topology sequence. The course covers the fundamentals of "point-set" topology. Topics include: the fundamentals of metric and topological spaces, continuous maps and homeomorphisms, connectness and path-connectedness, compactness, sub/product/quotient-spaces, separation axioms, countability axioms, and (time-permitting) metrization.
This course sits somewhere between two areas: (i) topology as the study of shapes (a coffee cup is the same as a donut, etc.), which is covered in greater detail in 190B; and (ii) topology as a huge and pleasing generalization of real analysis (what it means for a function to be continuous, the intermediate value theorem, etc.). The material covered is foundational and fundamental for both these and other areas.
There is no required textbook for the course. Part I of Topology by James Munkres covers some similar material and has been used in past versions of this course; it is suggested as-is, with no implied warranty, if you want an alternative exposition.
Contacts
The instructor is Freddie Manners (email fmanners; office AP&M 7343, not that that matters). The TA is Itai Maimon (imaimon). [All emails are at ucsd.edu.]
Class, section and office hours
Lectures are held on Mondays, Wednesdays and Fridays, 1100–1150. Lectures will be held on Zoom, with links in Canvas as usual.
Section will also take place via Zoom, on Wednesdays, 1800–1850.
See the calendar below for section times and office hours.
Virtual attendance in class is mandatory, and in-class activities carry credit. The only exception is if your timezone or circumstances make this logistically difficult. In that case, you may arrange to attend the lecture "doubly virtually" at a pre-arranged time. You must contact the lecturer to arrange this option, or discuss other logistical difficulties.
Section attendance is also mandatory, and credit is assigned to attending at least 80% of section. However, with the TA's approval you may leave before the end of section and still count as attending.
Please note that all lectures, sections and office hours may be recorded in Zoom and reposted to the class.
Exams
There are two midterm exams, in-class, on January 25 and February 19. The final exam is on March 15, 1130—1430.
You must ensure you are free on these dates and time, unless (as discussed above) there is a good reason why you cannot be, in which case you must raise this with the instructor at the start of the quarter.
Homework
Homework will be set every week, due by 2359 each Sunday night. The first homework deadline is on Sunday January 10; the last on Sunday March 8. There will be some optional homework in week 10 as exam practice.
The first week and midterm weeks will have reduced homework loads, but still some homework.
Please note: while discussing homework problems in groups is permitted (and encouraged), your final written-up solutions must be written by you, by yourself, in your own words. If your homework appears to have been copied directly from another student (or another source) that may constitute an academic integrity issue. You also may not post homework questions or solicit answers on the internet.
Grading
Your combined grade for the course is calculated as follows.
First, your lowest homework score is dropped. Then, take 25% homework + 3% in-class activities + 2% section attendance + 20% midterm 1 + 20% midterm 2 + 30% final. The letter grade cut-offs will be at least as generous as the following table (but may be more generous). Separately, exam scores may be curved to adjust for difficulty.| A+ | A | A- | B+ | B | B- | C+ | C | C- | F |
|---|---|---|---|---|---|---|---|---|---|
| 97 | 93 | 90 | 87 | 83 | 80 | 77 | 73 | 70 | < 70 |
Resources
In addition to this website, the course has a Canvas page and a Gradescope page. The sign-up code for Gradescope is listed on the Canvas home page.
You should find all Zoom links for the course on the course Canvas page under the "Zoom LTI" tab.
Provisional schedule
The rough, provisional, subject-to-change course schedule is given below. Note Week 1 starts on Monday January 4, etc..
| Week | Topic |
|---|---|
| 1 | Metric spaces. Examples. Subspaces. Open sets. Limit points. |
| 2 | More on metric spaces. Completeness. Continuity. Definition of a topological space. |
| 3 | MLK Day. Bases. Convergence of sequences. Continuity (again). Hausdorff spaces. Review. |
| 4 | Midterm 1. Closures and interiors. Subspace topology, quotient topology, product topology. |
| 5 | Connectedness. Definitions and properties. |
| 6 | Path-connectedness. Connected components. |
| 7 | Presidents Day. Review. Midterm 2. |
| 8 | Compactness. Sequential compacteness. |
| 9 | The Baire Category Theorem and applications. Countability axioms. |
| 10 | Further topics: Urysohn's lemma; metrization. Review. |
Homework assignments
Those assignments that have not been created yet link to a placeholder.
| Week | Deadline | |
|---|---|---|
| 1 | Sunday January 10, 2359 | p1.pdf |
| 2 | Sunday January 17, 2359 | p2.pdf |
| 3 | Sunday January 24, 2359 | p3.pdf |
| 4 | Sunday January 31, 2359 | p4.pdf |
| 5 | Sunday February 7, 2359 | p5.pdf |
| 6 | Sunday February 14, 2359 | p6.pdf |
| 7 | Sunday February 21, 2359 | p7.pdf |
| 8 | Sunday February 28, 2359 | p8.pdf |
| 9 | Sunday March 7, 2359 | p9.pdf |
| 10 | Never (optional pset) | p10.pdf |
Office hours
Regular office hours and locations are listed in the table below. However, please check the calendar below for any one-off changes or cancellations.
| Instructor / TA | Location | Regular hours |
|---|---|---|
| Freddie Manners | Zoom | Wednesdays 2:30pm – 3:30pm, Fridays 12:00 noon – 13:00pm |
| Itai Maimon | Zoom | Thursdays 2:00pm – 4:00pm |