Cardinality of an uncountable union of countably infinite sets?

Question

Let $I$ be an uncountable directed set and $T = \{E_{\alpha}:\alpha \in I\}$ denote a collection of Countably infinite sets with $E_{\alpha} \subset E_{\beta}$ whenever $\alpha \le \beta$. Also $T$ is totally ordered with respect to set incusion. What will the cardinality of $\bigcup_{\alpha \in I} E_{\alpha}$ be?

Answer 1

Lemma. If $X$ is a family of countable sets which is well-ordered under $\subseteq$, then $|X|\leq\aleph_1$.

There are two options:

1. The cofinality of $T$ is countable. Namely, there is a countable set $T_0$, such that for any $E_\alpha\in T$ there is some $E_\beta\in T_0$ such that $E_\alpha\subseteq E_\beta$. Therefore $\bigcup T=\bigcup T_0$.

2. There is no such countable set. In that case using the above lemma the cofinality of $T$ must be $\aleph_1$. Therefore, $\bigcup T$ is equivalent to the union of $\aleph_1$ countable sets, so it must have cardinality $\aleph_1$.