Overview: Topology studies shapes and surfaces, sometimes in higher dimensions, along with the continuity properties of functions between two such shapes. We begin with set theory and foundations, and then proceed to topological spaces, metric spaces, connectedness, compactness, countability and separation axioms, completeness, and function spaces.
Here is the catalog course description, and we also refer students, especially graduate students, to the first year exam topics.
Goals: Students will become fluent with the main ideas and the language of topology, and will be able to communicate these ideas to others. Students will learn how to write rigorous mathematical proofs and how to construct counterexamples. A goal of the class is to teach the foundations of rigorous argument through proving claims built on axioms.
Syllabus: Here is the course syllabus.
Book: The course book is Topology by James R. Munkres, Second Edition. You are expected to read the relevant sections, and to come to class with questions.
Notes
Henry’s lecture notes are split by chapter (of Munkres):
Lecture notes for Chapter 1
Lecture notes for Chapter 2
Lecture notes for Chapter 3
Lecture notes for Chapter 4
Lecture notes for Chapter 7
Lecture notes for Chapter 8
Homework
The clarity of your solutions is as important as their correctness. Working in groups on homework and to study is encouraged! However, your submitted homework should be written up individually, in your own words, and without consulting anyone else’s written solutions of any form.
Homework 1 (LaTeX source) is due Wednesday, September 6.
Homework 2 (LaTeX source) is due Monday, September 18.
Homework 3 (LaTeX source) is due Wednesday, September 27.
Homework 4 (LaTeX source) is due Friday, October 20.
Homework 5 (LaTeX source) is due Wednesday, November 1.
Homework 6 (LaTeX source) is due Monday, November 20.
Homework 7 (LaTeX source) is due Monday, December 4.
Homework solutions are posted on Canvas.
Exams
The exams will be in-class. You will only be able to use your brain and a pen or pencil – no notes, books, or electronic devices. The exams will be comprehensive, except that Exam 2 will emphasize the material after Exam 1, and the Final will emphasize the material after Exam 2.
Here is Practice Exam 1.
Here is Exam 1.
Here is Practice Exam 2.
Here is Exam 2.
Here is a Practice Final.
Here is the Final.
Exam solutions are posted on Canvas.
Schedule
Chapter 1: Set theory and foundations (Weeks 1-2)
Chapter 2: Topological spaces, continuity, products, bases, metrics, quotients (Weeks 3-6)
Chapter 3: Connectedness and compactness (Weeks 7-9)
Chapter 4: Countability and separation axioms (Weeks 10-12)
Chapter 7/8: Completeness, function spaces, contraction mapping theorem (Weeks 13-15)
| Date | Class Topic | Remark |
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| Aug 23 | §1, 2, 3: Sets, functions, relations | |
| Aug 25 | §4, 5: Integers, reals, products | |
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| Aug 28 | §6, 7: Countability | |
| Aug 30 | Hurricane Idalia | |
| Sep 1 | §7: Countability | |
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| Sep 4 | Labor Day | |
| Sep 6 | §7, 9: Infinite sets, axiom of choice | HW1 due |
| Sep 8 | §10, 11, 12: Axiom of choice, topological spaces | |
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| Sep 11 | §13: Basis for a topology | |
| Sep 13 | §14: Order topology | |
| Sep 15 | §15: Product topology on X×Y | |
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| Sep 18 | §16: Subspace topology | HW2 due |
| Sep 20 | §17: Closed sets and limit points | |
| Sep 22 | §17, 18: Continuous functions | |
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| Sep 25 | §18: Continuous functions | |
| Sep 27 | §18: Continuous functions | HW3 due |
| Sep 29 | §19: Product topology | |
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| Oct 2 | Practice Exam 1 Review | |
| Oct 4 | Exam 1 | |
| Oct 6 | Homecoming | |
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| Oct 9 | §20: Metric topology | |
| Oct 11 | §20, 21: Metric topology | |
| Oct 13 | §22: Quotient topology | |
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| Oct 16 | §23: Connected spaces | |
| Oct 18 | §24, 25: Path connectedness, local connectedness | |
| Oct 20 | §26: Compact spaces | HW4 due |
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| Oct 23 | §26: Compact spaces | |
| Oct 25 | §27, 28: Limit point compactness | |
| Oct 27 | §29: Local compactness | |
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| Oct 30 | §30: Countability axioms | |
| Nov 1 | §31, 32: Separation axioms, normal spaces | HW5 due |
| Nov 3 | Henry at FSU Topology Conference | |
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| Nov 6 | Practice Exam 2 Review | |
| Nov 8 | Exam 2 | |
| Nov 10 | Veterans Day | |
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| Nov 13 | §32, 33: Normal spaces, Urysohn lemma | |
| Nov 15 | §43: Complete metric spaces | |
| Nov 16 | Dinner at Henry and Ewo’s! | Starting at 7pm |
| Nov 17 | §43: Complete metric spaces | |
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| Nov 20 | §44: A space-filling curve | HW6 due |
| Nov 22 | Thanksgiving | |
| Nov 24 | Thanksgiving | |
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| Nov 27 | §48: Baire spaces | |
| Nov 29 | §48, 50: Baire spaces | |
| Dec 1 | §50: Dimension theory | |
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| Dec 4 | §50: Dimension theory | HW7 due |
| Dec 6 | Practice Final Exam Review | |
| Dec 8 | Reading Day | |
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| Dec 14 | Final Exam, 7:30-9:30am | Little Hall 223 |