1. ## Sets of Reals

Prove that there exists a $X \in \mathcal{P} (\omega)$ with $\left| X \right|= 2^{\aleph_0}$ such that every member of $X$ is infinite, but $a \cap b$ is finite for all distinct $a,b \in X$.

Prove that for every $X \subset \mathbb{R}$ there are disjoint subsets $Y,Z$ such that $X = Y \cup Z$, $Y$ is countable, and for no pair $a,b \in \mathbb{R}$ is the set $\left\{ c \in Z : a < c < b \right\}$ nonempty and finite.

2. Originally Posted by syme.gabriel
Prove that there exists a $X \in \mathcal{P} (\omega)$ [I'll assume you mean $\color{red}X \subset \mathcal{P} (\omega)$.] with $\left| X \right|= 2^{\aleph_0}$ such that every member of $X$ is infinite, but $a \cap b$ is finite for all distinct $a,b \in X$.

Prove that for every $X \subset \mathbb{R}$ there are disjoint subsets $Y,Z$ such that $X = Y \cup Z$, $Y$ is countable, and for no pair $a,b \in \mathbb{R}$ is the set $\left\{ c \in Z : a < c < b \right\}$ nonempty and finite.
For the first one, let $f:\mathbb{Q}\rightarrow\omega$ be an enumeration of the rationals. For every irrational number x, let $(a^{(x)}_n)$ be a sequence of rationals converging to x, and let $S_x = \{f(a^{(x)}_n):n\geqslant1\}$. Then X will be the collection of all these sets $S_x$.

For the second one, don't you just take Y to be the set of isolated points in X, or am I missing something?