# Math4121 Lecture 17

## Continue on Last lecture

### Countability

#### Theorem: $\mathbb{R}$ is uncountable

We denote the cardinality of $\mathbb{N}$ be $\aleph_0$

We denote the cardinality of $\mathbb{R}$ be $\mathfrak{c}$

> Continuum Hypothesis:
>
> If there a cardinality between $\aleph_0$ and $\mathfrak{c}$

### Power set

#### Definition: Power set

Given a set $S$, the power set of $S$, denoted $\mathscr{P}(S)$ or $2^S$, is the collection of all subsets of $S$.

#### Theorem 3.10 (Cantor's Theorem)

Cardinality of $2^S$ is not equal to the cardinality of $S$.

<details>
<summary>Proof of Cantor's Theorem</summary>

Assume they have the same cardinality, then $\exists \psi: S \to 2^X$ which is one-to-one and onto. (this function returns a subset of $S$)

$$
T=\{a\in S:a\notin \psi (a)\}\subseteq S
$$

Thus, $\exists b\in S$ such that $\psi(b)=T$.

If $b\in T$, then by definition of $T$, $b \notin \psi(b)$, but $\psi(b) = T$, which is a contradiction. So $b\notin T$.

If $b \notin T$, then $b \in \psi(b)$, which is also a contradiction since $b\in T$. Therefore, $2^S$ cannot have the same cardinality as $S$.

</details>

### Back to Hankel's Conjecture

$$
T=\bigcup_{n=1}^\infty \left(a_n-\frac{\epsilon}{2^{n+1}},a_n+\frac{\epsilon}{2^{n+1}}\right)
$$

is small

What is the structure of $S=[0,1]\setminus T$? (or Sparse)

- Cardinality (countable)
- Topologically (not dense)
- Measure, for now meaning small or zero outer content.

## Chapter 4: Nowhere Dense Sets and the Problem with the Fundamental Theorem of Calculus

### Nowhere Dense Sets

#### Definition: Nowhere Dense Set

A set $S$ is **nowhere dense** if there are no open intervals in which $S$ is dense.

#### Corollary: A set is nowhere dense if and only if $S$ contains no open intervals

$S'$ contains no open intervals