Topology

Math 631-001

Spring 2023

The following table contains the schedule for the course. This page will be updated regularly throughout the semester.

Week Dates Sections in the Book
1 01/23 - 01/27 I. Topological Spaces
    1. What is Topology?
    2. Topological Spaces 12
2 01/30 - 02/03     3. Basis for a Topology 13
    4. Topology via Order and Products 14, 15
3 02/06 - 02/10     5. The Subspace Topology 16
    6. Closed Sets and Limit Points 17
    7. Limits of Sequences and Separation Axioms 17
4 02/13 - 02/17     8. The Metric Topology 20
II. Continuity of Functions
    1. Continuous Functions 18
5 02/20 - 02/24     2. Topology of Infinite Products 19
    3. Continuity in Metric Spaces 20, 21
6 02/27 - 03/03     4. The Quotient Topology 22
III. Connectedness and Compactness
    1. Connected Spaces 23
7 03/06 - 03/10     2. Connected Subspaces of the Real Line 24
03/09 Midterm Exam (5:55pm-7:10pm)
8 03/13 - 03/17 No class! (Spring Break)
9 03/20 - 03/24     3. Components and Local Connectedness 25
    4. Compact Spaces 26, 27
10 03/27 - 03/31     5. Products of Compact Spaces 26, 37
11 04/03 - 04/07     6. Compactness in the Reals and Metric Spaces 27, 28
    7. Local Compactness 29
12 04/10 - 04/14 IV. Countability and Separation Axioms
    1. The Countability Axioms 30
    2. More Separation Axioms 31, 32
    3. The Urysohn Lemma 33
    4. The Urysohn Metrization Theorem 34
    5. The Tietze Extension Theorem 35
13 04/17 - 04/21 V. Fundamental Group and Covering Spaces
    1. Homotopy of Paths 51
    2. Some Terminology from Group Theory 52
14 04/24 - 04/28     3. The Fundamental Group 52, 59, 60
    4. Covering Spaces and Liftings 53, 54
15 05/01 - 05/05     5. A Sampling of Fundamental Groups 54, 59, 60
    6. Higher Homotopy Groups and Then?
16 05/11 Final Exam (4:30pm-7:15pm)

For the course, I will draw material from the following books: