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-# Lecture 17
+# 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 $S$.
+
+Proof:
+
+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 $S$.
+
+EOP
+
+### 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
+