Intro to General Topology

Math 451, Winter 2026

Class info Lecture: TueThu 14:35–15:55 Classroom: TROTT 0070 Course outline Zoom link (email instructor for the passcode)

Instructor Anush Tserunyan Office: BURN 915 Office hours: TueThu 16:05–17:00

Lectures Lecture 3 – 2026.01.12 Lecture 2 – 2026.01.08 Lecture 1 – 2026.01.06

Homework Homework 1 (partial)

Notes and suggested texts

• A. TSERUNYAN, The Cantor–Schöder–Bernstein theorem [pdf]
• T. GAMELIN, R. GREENE, Introduction to Topology (2nd ed.)
• G. FOLLAND, Real Analysis: Modern Techniques and Their Applications, only Chapters 0 and 4
• I. KAPLANSKY, Set Theory and Metric Spaces (2nd ed.)

Assessment

• Maximum of (15% homework + 30% midterm + 55% final) and (15% homework + 85% final).
• There will be 6 homework assignments (one every two weeks) submitted on Crowdmark.
• The midterm will be held on February 17 in class. The date of the final will be determined later by the exam office.

Topics

1. Basic (naïve) set theory
1. Set operations and functions, Axiom of Choice
2. Equinumerocity, the Cantor–Schröder–Bernstein theorem, finite/infinite sets, Dedekind infinity, pigeonhole principle, countability
3. Cantor's powerset theorem and continuum
4. Equivalence relations and quotients
2. Metric spaces
5. Examples (ℝ^d, Cantor space, Baire space, uniform metric on B(X, ℝ)), subspaces, isometries and Lipschitz functions
6. Open/closed sets and convergence
7. Cauchy sequences, completeness and the shrinking balls criterion, completeness of ℝ and of B(X, ℝ) in the uniform metric, completion
8. Polish spaces and the perfect set theorem
3. Topological spaces
9. Examples (including Zariski topology and Furstenberg's topology on ℤ), subspace topology, metrizability
10. Convergence, interior and neighbourhood, boundary and closure
11. Generation and bases, second countability, Lindelöf property and countable sub-basis lemma
12. First countability, separability, their relationship with second countability
13. Limit of a function and local continuity, continuous functions and homeomorphisms, uniform continuity in metric spaces and extensions
14. Nets, limit of a function and continuity in terms of nets
15. Separation axioms
16. Connectedness and path connectedness, permanence under continuous functions, zero-dimensional spaces and separation of closed sets by clopen
4. Product and function spaces
17. Product topology, examples
18. The non-metrisability of uncountable products, countable products of metric spaces, separability of ℝ​​^ℝ
19. Spaces of continuous functions: uniform metric, compact-open topology
5. Compactness
20. Open covers and intersecting closed sets, Alexander's prebasis (subbase) theorem
21. Tichonoff's theorem
22. Sequential compactness vs compactness and first countability, compactness in terms of nets
23. Compactness for metric spaces via total boundedness and completeness
24. The Arzelá–Ascoli theorem (optional)
6. Other cool but optional topics
25. Baire measurability, Baire spaces and the Baire category theorem, the Kuratowski–Ulam theorem, applications
26. Extension theorems for continuous functions: Urysohn's lemma and Tietze extension
27. Ultrafilters and Stone topology