Extension of a set to a Perfect set

Can any countable set in $\mathbb{R}$ be extended to a perfect set $P$ in $\mathbb{R}$ such that $R-P$ is uncountable?

• I guess you can extend it to $\mathbb{R}$ – Yanko Jun 3 '17 at 9:51

Let $C$ be a countable subset of $\Bbb R$. By definition, a perfect set of a topological space is closed, so if $C$ is dense in $\Bbb R$ then it cannot be extended to a perfect set $P$ with non-empty complement. Conversely, if $C$ is not dense in $\Bbb R$ then there is a non-empty (and, hence, uncountable) open interval $U$ of $\Bbb R$, disjoint from $C$ and we may put $P=\Bbb R - U$.
$Q$ cant be extended as any perfect set containing it will be $R$.