MT 446: Topology
Fall 2017
Time/Location: MWF 1-1:50, Carney Hall 303
Instructor: John Baldwin
Email: john.baldwin@bc.edu
Office: 545 Maloney Hall
Office Hours: TBA
Website: This is the official course website. I recommend that you add this url to the bookmarks bar of your browser as you will be visiting the site often for HW assignments and other important information.
Course Description: Topology is one of the main branches of mathematics and also one of the largest faculty interests in the math department at BC. It is roughly "the study of shape." I encourage you to read the wikipedia description here. This course is a rigorous introduction to some of the most fundamental concepts in topology. We will spend the first two thirds of the course developing the basics of "point-set topology." During the last third, we will introduce the fundamental group and other ideas from "algebraic topology." Along the way, we will see some beautiful applications of these ideas. One of the purposes of this course is to serve as a bridge between courses like analysis and "Euclidean and non-Euclidean Geometry" and the first year graduate course in algebraic topology.
Pre-requisites: The official co-requisites for this course are MT310 and MT320 or the equivalent, though, strictly speaking, we will not need much material beyond what is taught in MT216. Analysis and algebra are listed as co-requisites primarily in order to scare away people who are not yet mathematically mature enough to take this course. Indeed, this course will be fast-paced, and will demand a lot of you in terms of mental effort and time. That said, I'm confident the effort will be worth it.
Textbook: The text for this course is "Topology" (2nd edition), by James Munkres. It is more comprehensive than we need, but is unmatched in its clarity. We will cover Chapters 1-3 and 9, with a few supplementary topics. A more detailed outline is given below. The numbers in parentheses indicate the corresponding chapter and sections numbers from the textbook.
Set Theory and Logic (1)
Fundamental Concepts (1)
Functions (2)
Relations (3)
Cartesian Products (5)
Topological Spaces and Continuous Functions (2)
Topological Spaces (12)
Basis for a Topology (13)
The Order Topology (14)
The Product Topology I (15)
The Subspace Topology (16)
Closed Sets and Limit Points (17)
Continuous Functions (18)
The Metric Topology (20)
The Quotient Topology (22)
Connectedness and Compactness (3)
Connected Spaces (23)
Connected Subspaces of the Real Line (24)
Compact Spaces (26)
Compact Subspaces of the Real Line (27)
Limit Point Compactness (28)*
The Fundamental Group (9)
Homotopy of Paths (51)
The Fundamental Group (52)
Covering Spaces (53)
The Fundamental Group of the Circle (54)
Retractions and Fixed Points (55)
The Fundamental Theorem of Algebra (56)
The Borsuk-Ulam Theorem (57)
Deformation Retracts and Homotopy Type (58)
Homework: Homework assignments will appear weekly on this website. You must turn in your assignment by the end of class on the day it is due. Late homework will not be accepted. Collaboration is encouraged although you must write up your work on your own; do not simply copy another's work. Failure to abide by this rule violates your academic integrity (see below) and will be punished rather severely. A lot of emphasis will be placed on careful mathematical reasoning and proof. As such, writing style counts as much as having the right answer. Your homework solutions must be written in complete sentences, and must be clear, concise, and easily readable. A poorly-written or illegible solution will not receive full credit.
On Participation and Office Hours: Participation is strongly encouraged. We will operate under the idea that there are no stupid questions. It is important that you nip any confusions or misunderstandings in the bud as soon as possible. The best way to do this is to ask questions if you're not sure whether you understand something. If you don't feel comfortable doing this during class (I often didn't as an undergrad), please come to office hours, send me an email, or schedule an appointment to talk.
Exams: There will be two in-class midterms, tentatively scheduled for Friday, October 6 and Friday, November 17. The final exam will be held on Wednesday, December 13. Exams must be taken as scheduled, except for documented illness or family emergency. Let me know as soon as possible if you must miss a midterm or reschedule the final.
Grading: Homework (25%), midterms (20% apiece), final (35%).
Academic Integrity: I don't expect issues on this front (right?), but just in case, here is a code of conduct and consequences.
Students with disabilities: Please let me know if you require special accommodations for documented health reasons. In particular, inform me well in advance for special considerations during examinations.