I study physics and have very limited background in set theory.
Let $\{U_{ij}\}_{i\in I,j\in J}$ be a family of sets. For each fixed $i$, $\{U_{ij}\}_{j\in J}$ is a countable family of sets. For each fixed j, $\{U_{ij}\}_{i\in I}$ is a countable family of sets. Is that true that the family of unions $$\{\cup_{i\in I}U_{ij}\}_{j\in J}$$ is countable also? If this is true, is it related with the axiom of choice?