Exercise 2.29

Let $V$ be the open set in question. We know that the collection $U$ consisting of all segments of the form $(r-\alpha ,r+\beta )$, $r,\alpha \in \mathbb{Q}$, $\alpha ,\beta \in {\mathbb{Q}}^{\text{+}}$, is a countable basis for $ℝ$.

Let $V$ be the subcollection of sets of $U$ that are contained in $V$. Let $I_x$ be the union of all sets $U\in V$ such that $U$ intersects a segment in $V$ that contains $x$. It is clear that $I_x$ is a segment, $I_x\subset V$, and that if $y\in V$ then either $I_x=I_y$ or they are disjoint.

The collection ${\{I_x\}}_{x\in V}$ covers $V$, and since $ℝ$ is separable, it can be reduced to a countable subcover $I$. It is then clear that $I$ has the desired properties.

Let $E$ be an open set in ${ℝ}^1$. Suppose $\alpha$ and $\beta$ (without loss of generality, assume $\alpha <\beta$) are in the same segment if and only if $(\alpha ,\beta )\subset E$. Then every point of $E$ must lie in some segment, and all these segments are disjoint. Since ${ℝ}^1$ contains a countable dense subset (for example, ${\mathbb{Q}}^1$), we can choose one point from each segment that lies in this dense subset. This shows that the number of segments is at most countable.