MATH 440/640 (Fall 2015)

Instructor's contact information

Michael Robinson
226 Gray Hall
michaelr at american {dot} edu
Office hours:
My research website
Feel free to contact me with any and all questions (course-related or not)

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Course description
Homework assignments
Course schedule
Information about exams
Course policies

Course description

Topological spaces, continuity, compactness, connectedness, and metric spaces.

Prerequisite: MATH-403/603 Foundations of Mathematics (proving things about sets) or my permission. Conveniently, Chapter 0 of the textbook is a great resource for what you will typically need to remember!

The course textbook is Introduction to General Topology, by George L. Cain. Any edition will do.

In this course, students will

  1. Learn the definitions and key facts about topological spaces and continuous functions,
  2. Learn to classify spaces and functions according to topological properties,
  3. Develop an awareness of how topological ideas can be used outside of topology, and
  4. Continue to develop the skill of reading and writing mathematical proofs.


Homeworks are posted on BlackBoard, though I usually remember to print out copies to distribute in class.

Homework 1 due Friday, September 11
Homework 2 due Monday, October 5
Homework 3 due Monday, October 19
Homework 4 due Monday, November 2
Homework 5 due Monday, November 9
Homework 6 due Monday, November 23
Homework 7 due Monday, December 7 (at the final exam)

Course schedule

Unit 1: Topological spaces

After this unit, you should be able to
  1. Recite the definition of a topology and explain the topology of familiar spaces, such as the real line and the sphere
  2. Identify situations where non-Euclidean topologies are helpful
  3. Prove basic facts about how open and closed sets can be used to represent a topology

August 31: Sections 0.1-0.5: Sets, functions, relations, and the integers
September 2: Section 1.1: Pseudometrics
September 9: Section 1.2: Open and closed sets
September 14: Section 2.1: Topological spaces
September 16: Section 2.1: Topological spaces
September 21: Section 2.2: Topological bases
September 23: Section 2.3: Subspaces
September 28: Dr. Robinson at a conference. Now would be a good time to watch Video 1:
September 30: Dr. Robinson at another conference, for this project. Watch Video 2:
October 5: Section 2.3: Subspaces
October 7: Exam 1

Unit 2: Continuity

After this unit, you should be able to
  1. Recite both standard definitions of continuity for functions,
  2. Give examples and nonexamples of continuous functions between two topological spaces,
  3. Explain how continuous functions can be used to compare topologies, and
  4. Perform this comparison on some common topological spaces

October 12: Section 3.1: Continuity
October 14: Section 3.1: Continuity
October 19: Section 3.2: Homeomorphisms
October 21: Section 3.3: The weak topology
October 26: Section 10.1: The strong (quotient) topology
October 28: Exam 2

Unit 3: Topological properties

After this unit, you should be able to
  1. Recite and explain the definitions of connectness and compactness
  2. Explain why compactness has the name it does
  3. Explain how to use connectedness and compactness to distinguish two different topological spaces

November 2: Section 4.1: Connectedness
November 4: Section 4.1: Connectedness
November 9: Section 4.2: Connected components
November 11: Section 4.3: Path connectedness
November 16: Section 4.4: Local path connectedness
November 18: Section 5.1: Compactness
November 23: Section 5.1: Compactness
November 30: Section 5.2: One-point compactification
December 2: Review for the final

Final exam: Dec 7, 5:30pm-8:00pm

Course policies


For each exam, you will be permitted to bring one 3"x5" index card with handwritten notes on it. Electronic calculators will not be permitted. However, mechanical calculators (such as this or these) are fine. I am not exactly sure how such a tool would be useful on an exam, but if you figure out a good way, I'd be very interested.

Shortened semester

Every seven years or so, the Fall semester is 14 weeks long instead of the usual 15 weeks. This semester is one of those semesters. As per instructions from the University, faculty must "include additional instruction time -- beyond the classroom meetings". To that end, I would like you to watch the first two lectures (and optionally the third) in the DARPA Sheaf Tutorial, which I created recently: This will take you far afield of what's covered in this course, but will also give you a feeling for the extent of its applications outside mathematics. If you're interested, that's great! Please let me know, and I can set you up with a research project.

MATH 660 versus 440

Giving cogent technical talks is an important skill for all scholars. The sooner and more often you present and defend technical ideas in front of an audience, the better. As part of your training, if you're taking MATH 660 (the graduate version of the class), you must give one of the class lectures. If you're taking MATH 440 instead, you have the option to present one of the lectures but are not required. Here are the parameters for your presentation:
  1. Your presentation will be graded solely on whether you have done it or not.
  2. You must work with me to select which lecture you'll give (see the schedule above), since I have several lectures that I would like to give myself.
  3. You may present the material from your own notes or the book. It's your choice. (A word of advice: the proofs are important, but enlightening examples are much more important! Spend time gathering or creating examples. Fully understand them in advance.)
  4. Since the class is highly interactive -- even energetic -- you'll have to be ready defend whatever you present. I'll help you if you get stuck -- so you should not feel apprehensive -- but expect to get questions from the audience.
  5. You should expect to present for about 20 minutes, since I do not expect you to run the entire class period. After you've introduced the material, I'll follow up with my own examples and additional detail.


Since the course is taught in a highly interactive manner, you are generally expected to attend and participate in all of the lectures. Please contact me in advance if you cannot attend, especially in the case of an exam.


The course grade will be determined as follows:
25% Homework
25% Exam 1
25% Exam 2
25% Final exam

Late homeworks will not be accepted, unless there is an official (University-approved) reason for doing so.

Academic dishonesty

Academic dishonesty is a serious offense. As a start, you should read and understand our University's policies.

There is to be no collaboration with other people using any means during an exam. Doing so risks starting academic dishonesty proceedings. However, you may ask me questions during an exam, which I may answer at my discretion.

You may collaborate with other students in this class on homeworks, but the work you turn in must be your own. You may not collaborate with others outside of the class on homeworks without my express permission.

At a minimum, honesty consists of presenting your ideas clearly and in your own words, possibly orally. On the other hand, the creation and writing of proofs, examples, or counterexamples is a creative process. Here are a few typical cases that are relevant for homework:

  1. If you happen to create an example, counterexample, or proof for your work on your own, you need not notate it as such. (It is expected that you will rediscover standard examples, and these will not cause any concern.)
  2. If a colleague shows you an example, counterexample, or proof that you like, please credit that person by name in your write-up. You should expect that I may challenge you to explain your writing in your own words, possibly orally.
  3. If you write an example, counterexample, or proof as part of a group, please credit all members of your group by name in your write-up. You should expect that I may challenge you to explain your writing in your own words, possibly orally.

The mathematics community has a unique perspective on academic honesty and priority. The mathematics community has strict social guidelines for assigning credit, which I expect you'll adhere to. Due to the canonicity of certain mathematical results and constructions, sometimes well-intentioned people end up in priority disputes when they independently discover something. If you have any questions on this matter, you are expected to consult me directly for advice.