## Math 4171 – Topology I (Fall 2020)

This syllabus is a work in progress! I may add more details as we get closer to the beginning of classes.

### Section information

• Class time and location: Tuesdays & Thurdays from 10:00AM to 11:20AM in Brown 100 and online (hybrid)
• Office hours: Fridays from 5:00PM to 7:00PM (and Thursdays during class time.)

### Subject

This is an introduction to the fundamental ideas and results in Point Set Topology, with an emphasis on metric spaces. (A “space” is simply a set with additional mathematical structure provided to it. Thus, in mathematics, we talk about “vector spaces,” “measure spaces,” “topological spaces,” etc.) The language of topology pervades much of modern mathematics and some of the theorems we are going to learn are required for the more advanced courses in both pure and applied mathematics you are likely to take in the near future. Among the main topics, we will study: metric and topological spaces, product and quotient spaces, the general notions of continuity and convergence, connectedness, compactness, the Hausdorff property, Baire category, and the various theorems related to these notions. One way to think about this first course in Topology is that it extends certain notions that are familiar from calculus into much more general settings. This generality is important, particularly in areas such as mathematical analysis, where we need to apply concepts like convergence and continuity to infinitely dimensional function spaces, or spaces that are less familiar than the n-dimensional coordinate space of calculus. This course is also the foundation for Topology II, where you will explore the global structure of topological spaces with the aid of algebraic methods.

### Text

A Course in Point Set Topology by John B. Conway, Spring, 2014.

### Course plan

Given the exceptional circumstances in which we are all living, we’ll have to be flexible about plans and how we implement them (both on the instructor’s and on the students’ side). But the basic idea is this:

• The course will be structured around weekly homework sets. I expect we will have no more than 11 of them. (The one with the lowest grade will be dropped.) There is also one final exam which will be in most respects similar to a homework assignment (open notes and “take home”), except that it will be more inclusive in content. (I’ll call it the Final Homework. It cannot be dropped.) I plan to have the assignments available over the weekend, with a deadline set for late Friday; solutions will be available sometime over the weekend. (Although I won’t set penalties for late assignments, they won’t be accepted at all once the solutions are posted.) Assignments will be submitted and graded through Crowdmark.

• I plan to have lectures about the topic of the week recorded for (asynchronous) viewing relatively early in the week. In these lectures I’ll point out the parts of the textbook that are required reading (from John Conway’s text). On Tuesdays’ class, we will have discussions (jointly with the in person students and those attending via Zoom). These discussions will be about the content of the recorded lectures and the textbook’s assigned reading, with a particular focus on what is needed for the week’s homework assignment. As of now, I do not intend to record class meetings (due to privacy concerns), but this may change if there is strong preference for it.

• On Thursdays class, we will have “office hours” style meeting, where I’ll be available in person and via Zoom to discuss the homework (or anything else of interest to you!). Additional office hours will also be scheduled.

• This scheme is very new to me, and likely to you as well. It is likely that we’ll modify our routine somewhat as we learn what works most effectively and what doesn’t. I’ll also say more later about contingent plans if we are forced to move into a fully online scheme. And, of course, we’ll do our best to accommodate individual issues that may arise. The main point to have in mind is that your most important component of the course are the weekly assignments.

• Weekly homework assignments: no more than 11; the lowest score will be dropped.

• A final assignment worth twice as much as a single regular homework. (Due January 7) It can’t be dropped.

Letter grade cut-offs will be as follows:

• A (-, plain, +): cumulative score in [90%, 100%]

• B (-, plain, +): cumulative score in [80%, 90%)

• C (-, plain, +): cumulative score in [65%, 80%)

• D: cumulative score in [50%, 65%)

• F: cumulative score less than 50%.

The cut-offs for the letter grade signs (-, plain, +) will be set at the very end of the course, when all the scores have been computed. They will be set so as to make the overall number of -, plain, + roughly equal. (This is not the same as saying that each letter interval will be subdivided into three subintervals of equal length!)

I may change these cut-off scores if I find it necessary, although no changes will be made that would result in a tougher scale than the above.

### Good writing

A quick browsing the textbook should make it clear that proofs in point set topology are typically not based on computations as in calculus or algebra, but have a more discursive quality. To be readable, your argument needs to be concise and to the point, making sure that the essential elements of the proof are mentioned and, at the same time, that they are not obscured under lots of relatively minor details. It is not always easy to strike a good balance, but the textbook itself offers a good model to follow. When reading the textbook, pay close attention to the style in addition, of course, to the mathematical content.

Keep in mind:

• Never submit your first draft! Once you are happy with your solutions or proofs, rewrite them in a clean and orderly way. I will ask the grader to take points from messy and difficult to read assignments.

• Write with empathy! Put yourself in the shoes of the reader. Is your writing so wordy that the main points of a proof get lost amid lots of trivial observations, or are you writing so little that the reader won’t find your explanations too helpful?

• Although not necessary, this may be a very good course in which to practice your Latex skills! (But you will not get extra credit if you do.)