I want to find a nonempty perfect set with no rationals.
Let $\{ q_n \}$ be an enumeration of the rationals such that $\mathbb{Q} = \{ q_1, q_2 , ..., q_n ,... \}$. Define the open intervals $I_n = (q_n - \delta_n , q_n + \delta_n)$. I want to find a sequence $\{ \delta_n \}$ so that the set $\displaystyle P = \mathbb{R} \setminus \bigcup_{n=1}^\infty I_n$ is perfect and contains no rationals. $P$ is clearly closed by the DeMorgan Laws, as $\displaystyle \mathbb{R} \setminus \bigcup_{n=1}^\infty I_n = \bigcap_{n=1}^\infty (\mathbb{R} \setminus I_n)$, and the intersection of a collection of closed sets is closed. We have that if $\displaystyle \sum_{n=1}^\infty \delta_n < \infty$, then $\displaystyle\mathbb{R} \setminus \bigcup_{n=1}^\infty I_n\neq \emptyset$.
How would we pick $\delta_n$ so that $P \subset P'$?