We have $O \subseteq \mathbb{R}$ is an open set. We define the connected component $[x]$ of $O$ by an equivalence relation ~ on $O$: $x$ ~ $y$ for $x, y \in O$ if and only if there is a nonempty, connected subset $F$ of $X$ so that $x, y \in F$.
First, I prove that this is indeed an equivalence relation. Then, for my proof, it only remains to show that the set $\mathbb{O}$ which I define as $\{[x] \subseteq O| [x] \text{ is a connected component of }O\}$ is at most countable.
An idea: Since each $[x]$ is connected, we have (via a theorem) that $\forall y,z\in [x], [y,z]\subseteq [x]$. My idea is that since $\mathbb{Q}$ is dense in $\mathbb{R}$, there exists a rational number in every such interval. Still, I don't know that $\bigcup [y,z]=[x]$. I'd like to solve this myself, so hints and nudges are appreciated more than full answers! (However, if I am way off base, that is also good to know.)