MATH 540 (Fall 2013)

Instructor's contact information

Michael Robinson
226 Gray Hall
michaelr at american {dot} edu
Office hours: Monday 1-2pm; Tuesday 6-6:30pm; Wednesday 9:30am-10:15am and 1-2pm; Thursday 9:30am-10:15am, 1-2pm, and 6-6:30pm, or by appointment (please contact me 24 hours in advance to make arrangements)
My research website
Feel free to contact me with any questions (course-related or not)

Quick Links

Course description
Homework assignments
Course schedule
Concerning programming in this course
Information about exams
Some useful links
Course policies

Course description

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

Prerequisite: MATH-503 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.

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 comfort in using topological ideas outside of topology, and
  4. Continue to develop the skill of reading and writing mathematical proofs.


Homeworks are posted on BlackBoard.

Homework 1 due Tuesday, September 10
Homework 2 due Tuesday, September 24
Homework 3 due Tuesday, October 8
Homework 4 due Tuesday, October 22
Homework 5 due Tuesday, November 7
Homework 6 due Tuesday, November 21
Homework 7 due Tuesday, December 5

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 27: Abstract simplicial complexes
August 29: Sections 0.1-0.5: Sets, functions, relations, and the integers
September 3: Section 1.1: Pseudometrics
September 5: Section 1.2: Open and closed sets
September 10: Section 2.1: Topological spaces
September 12: Section 2.1: Topological spaces
September 17: Section 2.2: Topological bases
September 19: Section 2.3: Subspaces
September 24: Section 2.3: Subspaces
September 26: 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 1: Section 3.1: Continuity
October 3: Section 3.1: Continuity
October 8: Section 3.2: Homeomorphisms
October 10: Homotopies
October 15: Section 3.3: The weak topology
October 17: Section 10.1: The strong (quotient) topology
October 22: CW complexes
October 24: 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

October 29: Section 4.1: Connectedness
October 31: Section 4.1: Connectedness
November 5: Section 4.2: Connected components
November 7: Section 4.3: Path connectedness
November 12: Section 4.4: Local path connectedness
November 14: Section 5.1: Compactness
November 19: Section 5.1: Compactness
November 21: Section 5.2: One-point compactification
December 3: Applications!
December 5: Review for the final

Final exam: December 12, 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. That said, I have absolutely no idea how to prove a theorem using such a thing. If you can, I would be extremely interested!


You are generally expected to attend 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. Read our University policies. You may work together on homeworks, but the work you turn in must be your own.