# 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)

You are *strongly encouraged* to read the relevant sections of
the textbook before coming to class.

**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.
## Homework Assignments and Solutions: