Open sets of reals are countable unions of open intervals

Non-blind read-back — not independent testimony. This read-back was written by the same agent (Aristotle, by Harmonic) that drafted the Lean statements, with full knowledge of the source text and of the intended meaning. It was not produced by a blind, independent auditor, so it must not be mistaken for independent testimony; reviewers should compare the Lean code against the source themselves.

Order notions. For a (pre)ordered set $Q$: $Q$ is sparse if every closed interval $[a,b]=\{q: a\le q\le b\}$ is finite; $Q$ is dense (book sense) if $[a,b]$ is infinite whenever $a<b$; $A\subseteq Q$ is a dense subset if for all $q_1<q_2$ there is $q\in A$ with $q_1\le q\le q_2$; $Q$ is complete if every nonempty subset that is bounded above has a least upper bound in $Q$. A linear order $\preceq$ on $X$ is a natural order if the order topology of $\preceq$ (generated by all rays $\{x: a\prec x\}$ and $\{x: x\prec b\}$) equals the natural topology $\mathcal T$. $D$ is fully characterized by a topological space $Q$ if there is a homeomorphism $(X,\mathcal T)\to Q$. "Decidable" means $\Omega\setminus s\in\mathcal D$ for all $s\in\mathcal D$.

Statement. For every $U\subseteq\mathbb R$ that is open in the topology generated by the open intervals with rational endpoints, there are sequences $(a_i)_{i\in\mathbb N},(b_i)_{i\in\mathbb N}$ of extended reals ($[-\infty,+\infty]$) with $U=\bigcup_{i\in\mathbb N}\{x\in\mathbb R: a_i<x<b_i\}$. (Disjointness of the intervals is not asserted; an empty interval is allowed.)