Final guide

Five out of the six final questions will be slight variations of problems from this list.

PS1Q2 means Question 2 from Problem Set 1. T1Q2 refers to Question 2 from Tutorial 1.

1. Foundations

• PS1Q2. An example of a variation could be changing the word finite to countable.
• PS1Q9.

2. The Integers

• T4Q1 is part of what I consider basic properties of divisibility.
• PS2Q6 and PS2Q7.
• PS2Q8 is part of what I consider basic modular arithmetic.
• Last two exercises of 2.3.
• Last two exercises of 2.4.

3. Infinity

• PS3Q1, PS3Q6 and PS3Q11.
• Last exercise of 3.4.
• Second exercise and last exercise of 3.6.

4. Combinatorics

• There will be at least one question that uses the pigeonhole principle. Examples: T7Q1, PS4Q4 and PS4Q5, PS4Q6, T8Q3 and the exercises at the end of 4.2. Ignore the hard exercises.
• Any proof or exercise from the section Connectivity and trees except the hard or very hard ones.

5. Topology

• PS5Q1 is part of what I consider basic properties of open sets.
• The exercise in Interior and closure operators is part of what I consider basic properties of the interior and closure; also PS5Q9 and PS5Q10 have some of the same questions.
• Any union of open sets can be written as a countable union of open sets, even open balls.
• Any collection of pairwise disjoint nonempty open sets in $\mathbb{R}$ is countable.
• In PS5 I defined the sets $A^{\prime }$. Give an example of a set $A$ such that $(A^{\prime }{)}^{\prime }$ is a point. Can you find one where $((A^{\prime }{)}^{\prime }{)}^{\prime }$ is a point? What about $((((((A{)}^{\prime }{)}^{\prime }{)}^{\prime }{)}^{\prime }{)}^{\prime }{)}^{\prime }$?

6. Complex Numbers

• Only basic properties of complex numbers seen in the notes/PS6 excluding anything topological (like PS6Q4).