Given $I$ an index set and $\{A_i\}_{i \in I}$ where $\#A_i > 0$ and $A_i \cap A_j = \emptyset, i \neq j$ I is countable

Question

I am struggling to prove that given $I$ an index set and let $\{A_i\}_{i \in I}$ be a family of intervals where $\#A_i > 0 \ \forall i \in I$ and $A_i \cap A_j = \emptyset$ for $i \neq j$ then $I$ is countable. What would be an injective function $f: I \rightarrow \mathbb{N}$?

Answer 1

In every open interval there is a rational number.

Let $I \to \Bbb Q$ be a function which chooses a rational number from the interval $A_i$. This function is injective because the intervals are pairwise disjoint. Hence $I$ is countable.