Claim. Let $A_k$ be a countable set for each real number $k$. Then $\bigcup_{k \in \mathbb{R}} A_k$ is countable. Prove or Disprove

Question

CounterExample:

Let $A_k = \{k\}$, then Union of all $k$ $= \mathbb{R}$, and $\mathbb{R}$ is uncountable.

Is this a valid counter example?

Answer 1

Yes, it is a valid counter example. $\{ k \}$ is countable but $\mathbb{R}$ is not countable.

There is a result that countable union of countable set is countable but in this case the index set is not countable.