Let $C$ be a chain of countable sets, i.e. $\forall S, T \in C: S \subseteq T \lor T \subseteq S$, and that every $S \in C$ is countable. Then, is $\bigcup C := \{ t \mid \exists S \in C: t \in S\}$ countable? The answer is no, and a counter-example is $C = \omega_1$, the first uncounta...