The Bend in the River

Read-along: Rudin's PoMA: Week 3 (15th April to 21th April)

Visit the index page for this readalong

Goal: The goal for this week is to revise the relevant concepts of set theory that will be useful in our study of the topology of $ \mathbb{R} $.

Reading: The first section of chapter 2 of Rudin: more precisely, paragraphs 2.1 to 2.14 inclusive. The chapter is scanned here. An alternative treatment of the relevant material in set theory is the first chapter of Munkres’ Topology (in particular, sections 1.1 to 1.7); this chapter is scanned here. A classic introductory text to set theory is Paul Halmos’ Naive Set Theory (available online through the library).

In addition, there exists a joke about this material.

Suggested problems: Exercises 1.1 to 1.4.

It would be nice if everyone wrote up their solution to one or two of these problems so that we can collect them all online here and learn from each other ($ \LaTeX $ not required, just scan it if you want to as long as it’s readable!). If we end up having a seminar meeting this week then these solutions are something we can discuss there.

Additional exercises:

1. Let $ A $ and $ B $ be sets; define $ A^B := \{ f : f \text{ a function } B \to A \} $.
   • Define $ [n] := \{1,2,3,…,n-1\} $ for all $ n \in \mathbb{N} $. If $ A $ is a finite set, we may write $ \left| A \right| = n $ if $ A \sim [n] $. Show that if $ A $ and $ B $ are finite, then $ \left| A^B \right| = \left|A\right| \left|B\right| $.
   • Let $ X = \{0,1\} $. Show that there exists a bijection between $ \mathcal{P}(\mathbb{N}) $ and $ X^\mathbb{N} $.
2. Determine whether the following are countable:
   • The set of all functions $ f : \mathbb{N} \to \mathbb{N} $.
   • The set of all functions $ f : \mathbb{N} \to \{0,1\} $ such that there exists $ N \in \mathbb{N} $ making the implication $ [n > N \implies f(n) = 0] $ true.
   • The set of all finite subsets of $ \mathbb{N} $.
   • The set of all two-element subsets of $ \mathbb{N} $.
3. Theorem (Cantor). For any set $ X $, it is not the case that $ X \sim \mathcal{P}(X) $. [Hint: if $ f $ is an injection $ X \to \mathcal{P}(X) $, consider $ \{x \in X : x \notin f(x) \} $.]
4. Theorem (Schroeder-Bernstein). Let $ A $ and $ C $ be sets. If there exist injections $ f : A \to C $ and $ g : C \to A $, then $ A \sim C $.

Next: Next week (the second week of the mid-semester break) we will finally define metric spaces and their associated topology. I hope to also organise a physical meetup to discuss the exercises from weeks 1 to 3.