226 Gray Hall

michaelr at american {dot} edu

Office hours:

- Mondays 10am-noon, 4:15pm-5:15pm,
- Wednesdays 10am-noon,
- Thursdays 10am-noon, 1-2pm,
- or by appointment (please contact me 24 hours in advance to make arrangements).

My research website http://www.drmichaelrobinson.net/

Feel free to contact me with any and all questions (course-related or not)

Homework assignments

Course schedule

Information about exams

Course policies

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

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

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)

- Recite the definition of a topology and explain the topology of familiar spaces, such as the real line and the sphere
- Identify situations where non-Euclidean topologies are helpful
- 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: https://www.youtube.com/watch?v=b1Wu8kTngoE.

September 30: Dr. Robinson at another conference, for this project. Watch Video 2: https://www.youtube.com/watch?v=G3rWz2LgzZY.

October 5: Section 2.3: Subspaces

October 7: Exam 1

- Recite both standard definitions of continuity for functions,
- Give examples and nonexamples of continuous functions between two topological spaces,
- Explain how continuous functions can be used to compare topologies, and
- 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

- Recite and explain the definitions of connectness and compactness
- Explain why compactness has the name it does
- 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 **

- Lecture 1: Sheaf Theory: the Mathematics of Data Fusion (video: https://www.youtube.com/watch?v=b1Wu8kTngoE)
- Lecture 2: What is Topology? (video: https://www.youtube.com/watch?v=G3rWz2LgzZY)
- Lecture 3: What is a Sheaf? (video: https://www.youtube.com/watch?v=65y4UyjtPcM)

- Your presentation will be graded solely on whether you have done it or not.
- 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.
- 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.)
- 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.
- 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.

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.

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:

- 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.)
- 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.
- 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.