MATH 5345H Honors: Introduction to Topology (Fall 2019) |

Lecturer: Craig Westerland, 459 Vincent Hall, 612-625-0523, cwesterl@umn.edu.

Lecture: 11:15 -- 12:05 Monday, Wednesday, Friday, Vincent Hall 6.

Office Hours: 1:30 -- 2:30 Monday, 12:30 -- 1:30 Thursday, and 12:30 -- 1:30 Friday, Vincent Hall 459.

We'll learn to work with abstract topological spaces, both the concrete and the very formal, the non-intuitive and the geometric. We will develop qualitative tools to characterize them (e.g., connectedness, compactness, second countable, Hausdorff...), and develop tools to identify when two are equivalent (homeomorphic). Several important results will be proved, notably the Tychonoff theorem on products, but an equal focus will be placed on understanding examples coming from geometry, algebra, and number theory. Towards the end of the class, we will study the fundamental group and if time permits, covering spaces.

Through this course, students will learn to develop formal proofs and careful mathematical arguments, and will be able to communicate them effectively in writing. They will have mastered the basics of point-set topology and have a good understanding of the examples and counterexamples that inform the development of the subject.

- Set theory and Logic
- Topological spaces and continuous functions
- Connectedness and Compactness

- Countability and Separation axioms
- The Tychonoff Theorem
- Complete Metric Spaces and Function Spaces

- Baire Spaces and Dimension Theory
- Fundamental group, Seifert-van Kampen theorem
- Covering spaces

- J. R. Munkres, Topology, 2nd edition.

- (50%) Weekly homework assignments.
- (20%) Midterm takehome exam (18 -- 25 October).
**This must be handed in during class on 25 October, or emailed to the instructor by the start of class.** - (30%) Final takehome exam (due 5pm Monday, December 16).

Homework assignments:

- First Homework (due 13 September)
- Second Homework (due 20 September)
- Third Homework (due 27 September)
- Fourth Homework (due 4 October)
- Fifth Homework (due 14 October)
- Sixth Homework (due 18 October)
- Seventh Homework (due 8 November)
- Eighth Homework (due 18 November)
- Ninth Homework (due 27 November)
- Tenth Homework (due 11 December)