Question about the cardinality

Question

Let Let I denote the collection of all nonempty open intervals in R with rational endpoints, $$I = \{(a, b) : a, b \in\mathbb Q, a < b\}$$ What is the cardinality of I?

Let O denote the collection of all countable unions of open intervals in I. What is the cardinality of O?

Answer 1

$\aleph_0$.

You can identify each interval in $I$ (bijectively) by its pair of endpoints $\{(q_1,q_2)\in\mathbb{Q}^2\mid q_1<q_2\}$. Then use $|I|=|\mathbb{Q}^2| = |\mathbb{Q}|=|\mathbb{N}| = \aleph_0$.