We know that $\Bbb R$ is separable i.e. it contains a dense subset which is countable. We have $\Bbb Q$ and ${\Bbb R} - {\Bbb Q}$ to be dense subsets respectively countable and uncountable. I was looking for a countable dense subset of $\Bbb R$ which is a proper subset of either (i) $\Bbb Q$ or (ii) ${\Bbb R} - {\Bbb Q}$ .
For (i), by considering the set of dyadic rationals i.e. $\{\frac{a}{2^b} | a \in \Bbb Z , b \in \Bbb N \}$ or more generally for any fixed prime $p \in \Bbb N$, consider, $\{\frac{a}{p^b} | a \in \Bbb Z , b \in \Bbb N \}$ . It is a countable proper subset of $\Bbb Q$ which is dense.
But I could not come up with any example for (ii) . Thanks in advance for help.
Choose any rational number $a$. Then $\mathbb{Q}-\{a\}$ is a proper, dense subset of $\mathbb{Q}$.
Choose any irrational number $b$. Then $\{a+b \mid a \in \mathbb{Q}\}$ is a countable, dense subset of $\mathbb{R}-\mathbb{Q}$.