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 or, more generally, of cardinality less than $\kappa$ in part (c) (this only works for regular cardinals, so ignore this last general variation).
• 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 one question that uses the pigeonhole principle. To understand how to use this, try solving these by yourself: T7Q1, PS4Q4 and PS4Q5, PS4Q6 (which involves graphs), 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. Give an example of a set $A$ such that $A^{\prime \times 17}$ is a point. Now repeat the exercise where $A$ is a subset of the Cantor set.

6. Complex Numbers

• Any exercise from PS6.