Math 190A (Fall 2023)
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). The TA is Jordan Benson (jobenson).
Class and office hours
Lectures are held on Mondays, Wednesdays and Fridays, 10:00am–10:50am. They will be held in-person in APM 2402. Attendance is "mandatory but not enforced": if you choose not to attend, you will not lose points, but I also do not guarantee you will learn things.
Note there is no class on Wednesday November 22 (because of Thanksgiving).
Section will take place in-person at the advertized times and places.
The Google calendar below has times and locations for all class events.
Exams
There will be two (in-person, evening) midterms, on Wednesday October 25 (Week 4) at 7:00pm–8:20pm, in APM 2301, and Wednesday November 15 (Week 7) at 7:00pm–8:20pm in APM 2301.
The final exam is on Friday December 15, at 08:00am–11:00am. It is also in-person (location TBD).
You should ensure at the start of the quarter that you can attend these exams. In general, alternative modalities, dates or times (in particular, remote exams) will not be offered.
Homework
Homework will be set every week, due by 2359 each Wednesday night (with the exception of Week 4 and Week 7 when it becomes Thursday to accommodate the midterm). The first homework deadline is on Wednesday October 11 (Week 2); the last on Wednesday December 6 (Week 10). There will be some optional homework in week 10 as exam practice.
Midterm weeks and Thanksgiving Week will have reduced homework loads and in some cases altered deadlines, 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 is very likely to 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 + 20% midterm 1 + 20% midterm 2 + 35% final exam. 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, a Gradescope page and a Piazza site. The sign-up codes for Gradescope and Piazza are listed on the Canvas home page.
Provisional schedule
The rough, provisional, subject-to-change course schedule is given below. Note Week 1 starts on Monday January 4, etc..
| Week | Mon | Wed | Fri | Topic |
|---|---|---|---|---|
| 0 | L1 | Introduction | ||
| 1 | L2 | L3 | L4 | Metric spaces. Examples. Subspaces; open sets; completeness; continuity. |
| 2 | L5 | L6 | L7 | Definition of a topological space. Bases. Convergence of sequences. |
| 3 | L8 | L9 | L10 | Continuity. Hausdorff spaces. Closures and interiors. Subspace topology. |
| 4 | Review | L11 | L12 | Quotient topology, product topology. |
| 5 | L13 | L14 | L15 | Connectedness. Path-connectedness. |
| 6 | L15 | L16 | XXX | Connected components. Compactness. |
| 7 | Review | L17 | L18 | Sequential compactness. The Baire Category theorem. . |
| 8 | L18 | XXX | XXX | Applications of the Baire Category theorem. |
| 9 | L21 | L22 | L23 | Countability axioms. Urysohn's lemma. |
| 10 | L24 | L25 | Review | Metrization theorems; further topics (if time). |
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 | APM 7343 | Tuesdays 2:00–3:00pm, Wednesdays 12:30–1:30pm |
| Jordan Benson | HSS 4012 | Mondays 5:00–6:30pm, Wednesdays 3:00–4:30pm |