Prove "Any open subset of $\Bbb R$ is an at most countable union of pairwise disjoint open intervals”. Let $E$ be a nonempty open subset of $\Bbb R$. There is a collection $A$ of subsets of $\Bbb R$ satisfying the following:

$(i)$ Each element of $A$ is an open interval (possibly infinite) and any two distinct members of $A$ are disjoint;
$(ii)$ $A$ is at most countable;
$(iii)$ $E =\bigcup\limits_{G\in A}G$.


Prove that ”Any open subset of $\Bbb R$ is an at most countable union of pairwise disjoint open intervals”.


I don't know any structure theorems. I only have done a course on real analysis. Which dealt with compact sets and all.
Let $\mathcal{I}$ be the set of the intervals in $\mathbb{R}$ with rational endpoints and $E \subset \mathbb{R}$ an open set.

For all $x \in E$, prove there exist a rational $r_x &gt; 0$ and a rational $q_x$ such that $x \in\, ]r_x-q_x, q_x + r_x[$ and $]r_x-q_x, q_x + r_x[ \subset E$. Hint: use the density of $\mathbb{Q}$ in $\mathbb{R}$.
Deduce that $E$ can be written as a union of elements of $\mathcal{I}$.
Prove that $\mathcal{I}$ is countable. Hint: use the fact that $\mathbb{Q}\times \mathbb{Q}$ is countable.
Deduce you can write $E$ as a countable union of elements of $\mathcal{I}$.
Let $E = \cup_{\alpha \in A} I_{\alpha}$ be such a union. For each $\alpha \in A$, prove there exists a largest open interval $J_{\alpha}$ such that $I_{\alpha} \subset J_{\alpha} \subset E$. This $J_{\alpha}$ is not necessary in $\mathcal{I}$.
Conclude you can write $E$ as a countable union of pairwise disjoint intervals.
Let $\mathcal{I}$ be the set of the intervals in $\mathbb{R}$ with rational endpoints and $E \subset \mathbb{R}$ an open set.

For all $x \in E$, prove there exist a rational $r_x &gt; 0$ and a rational $q_x$ such that $x \in\, ]r_x-q_x, q_x + r_x[$ and $]r_x-q_x, q_x + r_x[ \subset E$. Hint: use the density of $\mathbb{Q}$ in $\mathbb{R}$.
Deduce that $E$ can be written as a union of elements of $\mathcal{I}$.
Prove that $\mathcal{I}$ is countable. Hint: use the fact that $\mathbb{Q}\times \mathbb{Q}$ is countable.
Deduce you can write $E$ as a countable union of elements of $\mathcal{I}$.
Let $E = \cup_{\alpha \in A} I_{\alpha}$ be such a union. For each $\alpha \in A$, prove there exists a largest open interval $J_{\alpha}$ such that $I_{\alpha} \subset J_{\alpha} \subset E$. This $J_{\alpha}$ is not necessary in $\mathcal{I}$.
Conclude you can write $E$ as a countable union of pairwise disjoint intervals.