A simple proof for: An open set in $\mathbb{R}$ is a countable union of open intervals

Question

I know that every open set in $\mathbb{R}$ is a countable disjoint union of open intervals (possibly infinite). It is not difficult to prove it. For example, there is a proof in Real Analysis (Folland). I am sure if we do not require intervals to be disjoint, there should be a much simpler proof. This is what I am looking for.

Answer 1

Let $G$ be an open set in ${\bf{R}}$. Let $x\in G$ and $r>0$ be such that $B_{2r}(x)\subseteq G$. Choose a rational number $p_{x}$ such that $|p_{x}-x|<r$. Choose a rational number $r_{x}>0$ such that $|p_{x}-x|<r_{x}<r$, then $x\in B_{r_{x}}(p_{x})$. Now we claim that $B_{r_{x}}(p_{x})\subseteq G$. For $y\in B_{r_{x}}(p_{x})$, then $|y-p_{x}|<r_{x}$, then $|y-x|\leq|y-p_{x}|+|p_{x}-x|<r_{x}+r<2r$, so $y\in B_{2r}(x)\subseteq G$, as expected.

So $G=\displaystyle\bigcup_{x\in G}B_{r_{x}}(p_{x})$ and all such $B_{r_{x}}(p_{x})$ are countably many.