[Math] Prove that the set of all open intervals with rational endpoints are countable.

Question

Prove that the set of all open intervals with rational endpoints are countable, I do not know exactly what shall I do, Could anyone help me please?

Answer 1

The set of rational numbers $\mathbb{Q}$ is countable. Furthermore, you should know that a finite product of countable sets is countable, so $\mathbb{Q}\times\mathbb{Q}$ is countable. (If you don't know this, prove it!) You can construct a bijection from your set to $\mathbb{Q}\times\mathbb{Q}$ the natural way, so that set will also be countable.