Math4121 Lecture 17

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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 $\mathfrak{c}$

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

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

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 $S$ .

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)

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

$S'$ contains no open intervals