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**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.