MTH 531/631: Topology I

University of Miami, Fall 2021

Instructor: Christopher Scaduto
Email: c.scaduto @ math.miami.edu
Office: Ungar 525
Office hours: Mondays 11:00 am - 12:30 pm (or by appointment)
Each office hour session is currently on Zoom.

Class Time and Location: 3:30-4:45 Mondays and Wednesdays in Dooly Memorial 209.

Course syllabus can be found here.

Text: Topology (2nd Edition) by James Munkres

The symbol § used below means "Section".

Course Schedule

Date Lecture Content Reading Notes
8/23/21 Introduction to the course. Set theory basics. Functions. § 1 and 2. Lec01
8/25/21 More on functions. Countable sets. § 2 and 7. Lec02
8/30/21 Uncountable sets. Topological spaces. § 7 and 12. Lec03
9/1/21 Bases of topologies. § 13. Lec04
9/6/21 Labor day (no class)
9/8/21 Subbases. Closed sets. § 13 and 17. Lec05
9/13/21 Limit points. Cantor set. § 17. Lec06
9/15/21 Convergence of sequences. Hausdorff spaces. Line with two origins. Product topology. § 17 and 15. Lec07
9/20/21 Subspace topology. Order topology. § 14 and 16. Lec08
9/22/21 More order topology. Metric topology. § 14 and 20. Lec09
9/27/21 More metric spaces. § 20 and 21. Lec10
9/29/21 Continuous functions. Homeomorphisms. § 18. Lec11
10/4/21 Continuous functions continued. § 18. Lec12
10/6/21 More on homeomorphisms. § 18. Lec13
10/11/21 Practice problems for Exam 1. Product topologies. § 19. Lec14
10/13/21 First exam (in class)    exam
10/18/21 Connectedness. § 23. Lec15
10/20/21 More connectedness. Path-connectedness. Compactness § 23, 24, 26. Lec16
10/25/21 More compactness. § 26. Lec17
10/27/21 Compactness again. § 27, 28. Lec18
11/1/21 Countability and Separation Axioms. Urysohn Lemma. § 30-33. Lec19
11/3/21 Urysohn Lemma continued. Urysohn Metrization Theorem. § 33, 34. Lec20
11/8/21 Topological manifolds. Partitions of unity. § 36. Lec21
11/10/21 Paracompactness. Quotient topologies. § 41, 22. Lec22
11/15/21 More Quotient spaces. Real projective space. Lec23
11/17/21 Manifolds with boundary. Classification of surfaces. reference Lec24
11/22/21 Break
11/24/21 Break
11/29/21 More on the classification of surfaces. Intro to fundamental group. Lec25
12/1/21 Second exam (in class)
12/6/21 More on the fundamental group.
12/8/21



Homework Assignments


Note: Problems graded are in blue. Apart these, remaining points are attributed to overall completeness of the assignment.
Assignment Due Date
Homework 1:    § 1: # 1 (Demorgan's Laws), 5, 10.    § 2: # 1, 2 (a-f) (b), 4 (c).     § 7: #3, 5 (a-f) (e)     9/7/21
Homework 2:    § 13: # 1, 3, 4 (a,c), 6, 7 9/13/21
Homework 3:    § 17: # 6 (a), 8, 11, 13, 19 (a, b)     9/22/21
Homework 4:    § 16: # 2, 3, 8, 9, § 17: # 2 9/29/21
Homework 5:    § 18: # 1, 8, 11, 12, § 20: # 3(a), § 21: # 10     10/8/21
Homework 6:    # 1, 2 (a) (b), 3, 4, 5 from here: hw6 10/27/21
Homework 7:    # 1, 2, 3, 4 from here: hw7 (try # 5 also!) 11/5/21
Homework 8:    # 1, 2, 3, 4 from here: hw8 11/12/21
Homework 9:    # 1, 2, 3 from here: hw9 11/19/21