MATH 5345H: Honors: Introduction to Topology — Fall 2026

Course Details

Class meetings:
- Time: 12:20 PM–2:15 PM, Monday and Wednesday
- Location: Vincent Hall 1
Credits: 4
Instructor:
- Name: Erkao Bao
- Office: Vincent Hall 356
- Email: bao@umn.edu
Office hours: To be announced on Canvas; additional meetings by appointment.
Course management: Announcements, assignments, submissions, and schedule updates will be posted on Canvas.
Textbook: Topology, 2nd edition, James R. Munkres. Any printing of the second edition is acceptable.

Course Description

MATH 5345H is a rigorous honors introduction to general topology. The course begins with set-theoretic and logical foundations and then develops topological and metric spaces, bases, subspaces, products, quotients, closed sets, limit points, continuity, and homeomorphisms. We will study connectedness, compactness, countability, and separation axioms, including selected major theorems such as Urysohn metrization, the Tietze extension theorem, and the Tychonoff theorem. The final portion introduces complete metric and function spaces and the fundamental group, with covering spaces and Seifert-van Kampen applications as time permits.

Equal attention will be given to abstract arguments and to examples and counterexamples arising from geometry, algebra, analysis, and number theory. Because this is an honors course, students should expect a fast pace, substantial independent proof writing, and regular engagement with nonstandard examples.

Prerequisites

One of MATH 2263, MATH 2374, or MATH 2573; and enrollment in or completion of MATH 2283, MATH 2574, or MATH 3283. Prior experience reading and writing proofs is required.

Course Goals and Objectives

Construct and compare topologies, and determine whether maps are continuous or are homeomorphisms.
Analyze how topological properties behave under subspaces, products, quotients, and continuous images.
Use connectedness, compactness, countability, and separation axioms to prove structural results.
Relate metric completeness, compactness, and convergence, including selected function-space results.
Define the fundamental group and compute it for basic spaces using coverings, deformation retracts, and van Kampen-type arguments.
Write precise, readable proofs and evaluate mathematical arguments for correctness, clarity, and completeness.

Mathematical Writing

Although this course is not officially writing intensive, mathematical writing is central to the course. A proof must be logically correct, but it must also present a convincing and readable argument to its audience. Homework and exams will be evaluated for organization, precision, appropriate use of notation, and explanation of the main ideas—not only for the final conclusion. Feedback will emphasize both mathematical correctness and exposition.

Course Format and Important Dates

The course meets in person twice each week and combines lecture, discussion, worked examples, and student proof development. Students should complete the listed reading before class. The final examination occurs outside the regular meeting time: Friday, December 18, 2026, 10:30 AM-12:30 PM. The final-exam modality and location will be confirmed on Canvas.

First class meeting: Wednesday, September 9, 2026
Midterm Exam 1: Wednesday, October 14, 2026, during class
Midterm Exam 2: Wednesday, November 18, 2026, during class
Thanksgiving holiday: Thursday–Friday, November 26–27, 2026; no scheduled MW class falls on these dates
Last day of instruction: Wednesday, December 16, 2026
Final exam: Friday, December 18, 2026, 10:30 AM–12:30 PM; modality and location TBA

Academic Support

Form study groups and use office hours for questions about concepts, proof strategy, and mathematical writing.
The Tutoring & Academic Success Center offers peer tutoring and related academic support.

Assessments

Homework

Problem sets will normally be posted weekly on Canvas and submitted through Canvas. A selected subset of problems may be graded. Unless announced otherwise, assignments are due by 11:59 PM on the date listed in the course schedule. The lowest problem-set score will be dropped. Extra credit is not intended to be part of the course.

Exams

Midterm Exam 1: October 14, 2026, 12:20 PM–2:15 PM, in class.
Midterm Exam 2: November 18, 2026, 12:20 PM–2:15 PM, in class.
Final Exam: December 18, 2026, 10:30 AM–12:30 PM. The exam is cumulative; modality and location will be confirmed on Canvas.

The exact format and permitted materials for each exam will be announced on Canvas. Collaboration is not allowed on exams.

Grading Policy

Homework: 30%
Midterm Exam 1: 20%
Midterm Exam 2: 20%
Final Exam: 30%

The standard weighted score and an alternate score that places additional weight on the stronger midterm and final exam will both be computed.

Score 1: Homework (30%) + Midterm 1 (20%) + Midterm 2 (20%) + Final Exam (30%)
Score 2: Homework (30%) + max(Midterm 1, Midterm 2) (30%) + Final Exam (40%)

The final numerical score is the larger of Score 1 and Score 2. The letter grade is the more favorable of a fixed scale (90: A range, 80: B range, 65: C range, 50: D range) and an approximate class distribution (top 25%: A range; next 30%: B range; next 35%: C range; remaining 10%: D/F). Plus and minus grades may be used. The curve will not lower a grade earned under the fixed scale.

Course Policies

Attendance

Attendance is not a separate graded component, but regular attendance and participation are expected. Students are responsible for announcements, assigned reading, and material covered during any absence. Excused absences and makeup work are handled in accordance with University policy.

Collaboration and Academic Integrity

Collaboration on homework is encouraged, but each student must write and submit an independent solution in their own words and must acknowledge collaborators and outside sources. Copying another solution, sharing completed write-ups, or submitting material produced by another person is not permitted. Generative AI tools are not permitted on exams and may be used for homework only when a particular assignment explicitly authorizes them; any authorized use must be disclosed. Collaboration is never permitted on exams.

Late Work and Make-up Exams

Late homework is accepted only for compelling reasons. Contact the instructor as soon as possible, preferably before the deadline. The dropped-problem-set policy is intended to absorb an ordinary missed or unusually weak assignment. Make-up exams are generally not allowed. A missed midterm may be accommodated only for a compelling excused circumstance and with prompt communication; advance permission should be obtained whenever possible.

Calculator and Electronic Device Policy

Calculators are not required and will not be permitted during exams. Personal electronic devices may be used during class only for course-related purposes and must not disrupt instruction or other students.

Incompletes

An incomplete will be considered only when substantial course work has been completed, the student is unable to finish for documented circumstances beyond their control, and a written completion agreement is made in accordance with University policy.

Drops and Withdrawals

Students are responsible for reviewing the University’s drop/add deadlines and withdrawal rules.

Course Schedule

The schedule is a good-faith plan and may be adjusted to match the pace and needs of the class. Changes will be announced on Canvas. The Fall 2026 term begins Tuesday, September 8; because this course meets Monday and Wednesday, the first meeting is Wednesday, September 9.

Week	Date	Day	Preparation	Topic	Homework
1	Sep 9	Wed	Syllabus; Munkres §§1-4	Course overview; proof writing; sets, functions, relations, and standard number systems	HW 1 assigned (due Sep 21)
2	Sep 14	Mon	Munkres §§5-7	Cartesian products; finite, countable, and uncountable sets	—
2	Sep 16	Wed	Munkres §§8-11	Recursive definition, choice principles, well-ordering, and maximality	—
3	Sep 21	Mon	Munkres §12	Topological spaces and standard examples	HW 1 due; HW 2 assigned
3	Sep 23	Wed	Munkres §13	Bases, subbases, and comparison of topologies	—
4	Sep 28	Mon	Munkres §§14-15	Order topology and finite-product topology	HW 2 due; HW 3 assigned
4	Sep 30	Wed	Munkres §§16-17	Subspaces, closed sets, closure, and limit points	—
5	Oct 5	Mon	Munkres §18	Continuity, homeomorphisms, and the pasting lemma	HW 3 due; HW 4 assigned
5	Oct 7	Wed	Munkres §19	Arbitrary products and coordinate maps	—
6	Oct 12	Mon	Munkres §§20-21	Metric topologies, metrizable spaces, and sequences	HW 4 due
6	Oct 14	Wed	Review Chapters 1-2	Midterm Exam 1	No homework
7	Oct 19	Mon	Munkres §22	Quotient topology and identification spaces	HW 5 assigned
7	Oct 21	Wed	Munkres §§23-24	Connectedness and connected subspaces of the real line	—
8	Oct 26	Mon	Munkres §25	Components, path components, and local connectedness	HW 5 due; HW 6 assigned
8	Oct 28	Wed	Munkres §26	Compactness and the finite-intersection property	—
9	Nov 2	Mon	Munkres §27	Compact subsets of Euclidean space; Heine-Borel and related results	HW 6 due; HW 7 assigned
9	Nov 4	Wed	Munkres §28	Limit-point compactness and sequential compactness	—
10	Nov 9	Mon	Munkres §29	Local compactness and one-point compactification	HW 7 due; HW 8 assigned
10	Nov 11	Wed	Munkres §§30-31	Countability and separation axioms; examples and counterexamples	—
11	Nov 16	Mon	Munkres §§32-33	Normal spaces and the Urysohn lemma	HW 8 due
11	Nov 18	Wed	Review §§22-33	Midterm Exam 2	No homework
12	Nov 23	Mon	Munkres §§34-35 (selected)	Urysohn metrization and the Tietze extension theorem	HW 9 assigned
12	Nov 25	Wed	Munkres §37 (selected)	The Tychonoff theorem and product compactness	—
13	Nov 30	Mon	Munkres §43	Complete metric spaces and completions	HW 9 due; HW 10 assigned
13	Dec 2	Wed	Munkres §§45-47 (selected)	Compactness in metric and function spaces; the Ascoli theorem	—
14	Dec 7	Mon	Munkres §§51-52	Path homotopy and the fundamental group	HW 10 due; HW 11 assigned
14	Dec 9	Wed	Munkres §§53-54	Covering spaces and the fundamental group of the circle	—
15	Dec 14	Mon	Munkres §§55, 58, 70	Retractions, fixed points, deformation retracts, and van Kampen setup	HW 11 due
15	Dec 16	Wed	Munkres §§71-73; §§79-82 if time	van Kampen applications; covering-space classification; course review	—
Finals	Dec 18	Fri	Review all course material	Cumulative final exam, 10:30 AM-12:30 PM; modality/location TBA	—

Required University Policy Statements

The University of Minnesota Recommended Policy Statements for Syllabi are incorporated by reference. They address the Student Conduct Code, personal electronic devices, scholastic dishonesty, excused absences and makeup work, appropriate use of class notes and course materials, University grading scales, sexual misconduct, equity and equal opportunity, disability accommodations, mental health and stress management, and academic freedom and responsibility.

Review the complete, current statements in the University’s Recommended Policy Statements for Syllabi.

Students seeking disability-related accommodations should contact the Disability Resource Center and share a current accommodation letter with the instructor as early as possible.