In Apostol, page $51$, he defines what he calls the component interval. I can't find any reference to it on the web. I have some problems with the definition:
Let $S \subseteq \mathbb{R}$. An open interval $I$ of $S$ is a component interval if there does not exist an open interval $J$ of $S$ such that $I \subset J$.
Intuitively, I get that $I$ is the largest possible open interval that is contained in $S$. I think that the set of all rationals between the end points of $S$, $\{\alpha \in (A,B)\ |\ \alpha \in \mathbb{Q}\}$, is a component interval. Is that true? If $D$ is dense in $S$, is $D$ in general a component interval of $S$?