Math 432: Set Theory and Topology

Spring 2019

Textbook: Kaplansky I, Set Theory and Metric Spaces, Second Edition, AMS Chelsea Publishing, 1957
Additional notes: A. Tserunyan, A quick introduction to basic set theory [pdf]
What will be covered:
- Informal/naïve set theory: sets, set operations, functions
- The Zermelo–Fraenkel (ZF) axioms
- Well-orderings and ordinals
- Transfinite induction
- The set of natural numbers
- Equinumerosity: finite/infinite sets, cardinals and cardinality
- The Axiom of Choice and equivalent statements
- Introduction of reals via Dedekind cuts
- Metric spaces: examples (including the Cantor space), open/closed sets, sequences, compactness, (uniform) continuity, completion
- General topological spaces

Coverage: Everything covered in the course (see the coverage of the midterms below), with emphasis on the material after Midterm 2: Dedekind cuts and completion of linear orders; metric spaces (including important examples); topological spaces (including examples); product topology; closure and boundary; convergence; continuity and sequential continuity; dense sets and separability; base, neighborhood base and first/second-countability; completeness of metric spaces and of the space of reals; compactness and sequential compactness.
Practice problems for topology: pdf
Review recommendation: Review lecture notes; redo Midterm 1 and Midterm 2 and check your solutions with mine; do many (ideally most) of the practice problems for topology posted above; look through homework, redo the problems whose solutions you don't recall; look through the practice problems for both of the midterms.
Jenna's help session: 12–1pm on Sun (May 5) in 147 Altgeld Hall.
Extra office hour: 12–1pm on Mon (May 6) in my office.

Homework will be assigned every week and it will consist of 6–10 problems. The submission of homework will be done as follows. The students will have to write up their solutions and submit their write-ups to Jenna Zomback in the weekly problem sessions, during which they will also be asked to present some of their solutions on the board. The file below describes exactly how the problem sessions work and the students are evaluated.
How the problem sessions work [pdf] — read this very carefully!
Lowest homework score we be dropped.

Grading scheme: total grade = homework 14% + midterms 2 x 23% + final exam 40%
Where to look for your grades: All of your scores for homework and exams will be recorded on Moodle (learn.illinois.edu). Log in with your UIUC netID and click on Math 432.
Letter grades: The final letter grade will be assigned based on the overall percentage obtained by the above grading scheme. The overall percentages will be curved in the very end of the semester, but no exam (midterms and final) will be curved individually.
Letter grade distribution: Up to minor adjustments, the top 37% of students will get an A grade (including plus and minus), the middle 35% a B grade, and the bottom 28% a C grade and lower.
Grading disputes: Regularly check your homework and exam scores on Moodle to verify that they are correct. Any grading complaints must be submitted within two weeks of the due date or exam date of the grade in question.

Coverage: ordinals, order type, transfinite induction, recursive definitions, the set of natural numbers, equinumerosity, finite vs. infinite, cardinals, usage of Axiom of Choice and friends (Zermelo, Zorn) without proofs of their equivalences.
What to read: Sections 3–8 of my basic set theory notes [pdf].
Practice problems: pdf
Extra office hour: 4–4:50pm on Wed (Mar 27) in my office.
Problem sessions this week: None, since no homework is due this week.

Coverage: sets, set operations, relations, functions, partial orderings, equivalence relations, formal language and formulas, ZFC axioms, well orderings.
What to read: Kaplansky's Sections 1.1–1.5 except for lattices in 1.3 and Sections 1–2 of my basic set theory notes [pdf].
Practice problems: pdf