Thm: Every nonempty open set $G$ of real numbers is the union of a finite or countable family of pairwise disjoint open intervals.
Proof: Let $x$ be a point in a nonempty open set $G$. There is an open interval $(y, z)$ such that $x\in (y, z)\subset G$. Then, $(y, x)\subset G$ and $(x, z)\subset G$. We define (possibly extended) numbers $a_x$ and $b_x$ by $a_x = inf (y : (y, x)\subset G)$ and $b_x = sup (z : (x, z)\subset G)$...
My question. Since $(y,x)$ and $(x,z)$ are intervals in G (hence, connected) wouldn't $a_x = b_x =x$?