MTH 531/631: Topology I

University of Miami, Fall 2022

Instructor: Christopher Scaduto
Email: c.scaduto @ math.miami.edu
Office: Ungar 525
Office hours: Tues/Thurs 11:30-12:30. I will also typically be available right after each lecture.

Class Time and Location: 12:30-1:45 Tuesdays and Thursdays in Ungar 411.

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
8/23/22 Introduction to the course. Set theory basics. Functions. § 1 and 2.
8/25/22 More functions. Cardinality. Countability. § 2 and 7.
8/30/22 More (un)countability. Cantor set. Continuum hypothesis. § 7.
9/1/22 Topological spaces. Topology of the reals. Bases. § 12, 13.
9/6/22 Bases continued. Finer/coarser. Product topology. § 13, 15
9/8/22 Subspace topology. Closed sets. Closure and interior. § 16, 17
9/13/22 Closures and interiors continued. Limit points. Dense sets. § 17
9/15/22 Hausdorff spaces. Line with two origins. § 17
9/20/22 Continuous maps. Homeomorphisms. (Zoom lecture) § 18 notes
9/22/22 Topological properties. Punctured sphere is homeomorphic to the plane. (Zoom lecture) § 18 notes
9/27/22 Metric spaces. § 20 notes
9/29/22 Practice problems. Metrizability. Sequence lemma. § 21 note
10/4/22 EXAM 1 Solns
10/6/22 Second countability. More homeomorphisms: "Polar coordinates". Square and disk.
10/11/22 Star-shaped set is homeomorphic to a ball. Quotient topologies. § 22
10/13/22 Break!
10/18/22 More on quotient topologies. The torus. Real projective spaces.
10/20/22 Some remarks on projective spaces. Connectedness. § 23, 24
10/25/22 Connected components. Disjoint unions (topological sums). § 24, 25
10/27/22 Path-connectedness and path-components. Topologists' Sine Curve. § 24, 25
11/1/22 Compactness. Closed intervals are compact. Continuous maps and compactness. § 26, 27
11/3/22 More compactness. Heine-Borel theorem. § 26, 27
11/8/22 Lebesgue number Lemma. Other versions of compactness. Product topology for infinite products. Tychonoff Theorem. § 27, 28, 37
11/10/22 Countability axioms. Separation axioms. § 30, 31, 32
11/15/22 Discussion of Urysohn Metrization Theorem. Proof of Urysohn Lemma. § 33
11/17/22 Proof of Urysohn Metrization Theorem. Intro to topological manifolds.
11/22/22 (Zoom lecture) Practice problems. Solns.
11/24/22 Thanksgiving Holiday
11/29/22 EXAM 2
12/1/22
12/6/22 Last class!



Homework Assignments



Assignment Due Date
Homework 1: hw01 9/1/22
Homework 2: hw02 9/13/22
Homework 3: hw03 9/22/22
Homework 4: hw04 9/29/22
Homework 5: hw05 10/20/22
Homework 6: hw06 11/1/22
Homework 7: hw07 11/15/22
Homework 8: hw08 11/23/22