1. ## Sets of Reals

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

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

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

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

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