Book

The main reference for the class is Topology: A Categorical Approach. Chapter pdfs are available for free: open access version. I'll be supplementing with other material.

Practical information

The class meets in room 5417 in the CUNY Graduate Center. It is scheduled to meet Mondays and Wednesdays from 2:00 -- 3:30. The first class is Monday, August 29.

Contact info:

• Office hours: Monday and Wednesday from 3:30 -- 4:30 in my office 4214.14 and by appointment
• email: jterilla@gc.cuny.edu

Videos

• What is topology? YouTube video and notes WhatIsTopology.pdf
• The subspace topology YouTube video and notes TheSubspaceTopology.pdf
• Paths YouTube video and notes Paths.pdf
• Compact and Hausdorff YouTube video.
• Filters and ultrafilters. Part I (background, no topology) YouTube video and Part II (topology, i.e. filter convergence and topological properties) YouTube video.
• Categorical limits (and colimits) are unique: YouTube video
• Compactifications YouTube video.
• Exponential topologies: YouTube video
• The Seifert Van Kampen Theorem YouTube video
• Covering spaces Part 1 Youtube video. Notes in pdf: Covering_Spaces_Part1.pdf.
• G-Sets YouTube video. Notes in pdf G-Sets.pdf.
• Covering Spaces Part 2 --- The Big Picture YouTube Video and notes in pdf CoveringSpacesPart2.pdf.

Homework

• Due Wednesday, September 7: Exercises 3abde, 4, 5 in Chapter 0. Also, examine the Yoneda Lemma when the category C is a group, thought of as a category with one object.
• Due Monday, September 19: Exercises 2, 9, 10, 14 in Chapter 1. Read Chapter 2.
• Due Friday, September 30: Exercises 14, 15, 16, 18, 19, 24 in Chapter 2.
• Due Wednesday, October 12: Watch the videos on filters and prove the following three theorems.
  1. Every ultrafilter on a finite set is principal (this is like saying a vector space is isomorphic to its dual iff it is finite dimensional. why?)
  2. A space X is compact iff every ultrafilter converges to at least one point.
  3. A space X is Hausdorff iff every ultrafilter converges to at most one point.
• Due Monday, October 17 and Wednesday, October 19: Read chapter 4 of the text and watch the video on the uniqueness of limits and colimits. Also, I pulled a few problems from past topology qualifying exams, let's make these problems due on Wednesday, October 19: homework5.pdf and LaTeX homework5.tex.
• Due Friday, November 11 Monday, November 14: homework6.pdf and LaTeX homework6.tex. As requested, I pulled these problems from old qualifying exams. Some of them I've solved in class. Most of them would make problems for an in-class exam.
• Here is some homework on covering spaces: homework7.pdf and LaTeX homework7.tex and cover4bw.pdf. Let me set the due date for this as Monday, December 5. You can have an extension, but you'll get the most out of Monday's class if you do this homework before then.
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