During Fall 2020, I taught MATH535 Topology, syllabus of which can be reached from here. Due to the COVID-19 pandemic, classes were held online and we had live lectures every week. These are the lecture notes of each live lecture.*
In order to prepare these notes, I mainly followed “Topology” by Munkres as the course textbook. However, I changed the order and extent of some topics. For certain topics, I also used “Topology” by Waldmann and introductory parts of “A course on Borel set” by Srivastava. I sometimes consulted to MathOverflow, MSE, ProofWiki, π-Base and many other sources that I found via Google searches, e.g. Mike Shulman’s notes for a talk, Pete Clark’s notes etc.
Use these lecture notes with caution, it is likely that there are many typos and errors. Please let me know of these if you spot any. Here are the notes of each lecture:
- Lecture 1: Topology and basis of a topology
- Lecture 2: Subbasis, closed sets and closures
- Lecture 3: Limit points, limits of sequences and some separation axioms
- Lecture 4/5: Order topology, Cantor-Bendixson rank and the Long Line
- Lecture 5/6/7: Cartesian products, box topology and product topology
- Lecture 7/8/9: Continuous maps and homeomorphisms, sequential continuity
- Lecture 9/10/11: Metric spaces and product metric
- Lecture 12/13: Completely metrizable spaces, Alexandrov’s theorem and uniform topology
- Lecture 14: Baire category theorem and some applications
- Lecture 15/16/17: Connected, totally disconnected, path connected spaces and the IVT
- Lecture 17/18: Compact spaces and sequentially compact spaces
- Lecture 19: Ultrafilters and Tychonoff’s theorem
- Lecture 20: Compact subsets of linearly ordered sets, extreme value theorem and compact metric spaces
- Lecture 21: Quotient topology
- Lecture 22/23: More on countability and separation axioms, normal spaces
- Lecture 24: Urysohn’s lemma
- Lecture 25: Urysohn’s metrization theorem and Tietze’s extension theorem
*: Due to privacy issues, I cannot provide the video recordings of lectures without obtaining permission from each student that attended the lectures, which I have not done.