Theorem: Every open set $\displaystyle U \subset \mathbb{R} $ can be uniquely expressed as a countable union of disjoint open intervals. The endpoints of $\displaystyle U $ do not belong to $\displaystyle U $ (Pugh, p. 63).

The book goes on to prove this by first defining $\displaystyle a_x = \inf\{ a : \exists(a,x) \subset U \} $ and $\displaystyle b_x = \sup \{b: \exists (x,b) \subset U \} $ and then an interval $\displaystyle I_x = (a_x, b_x) $.

I don't understand why they define $\displaystyle \sup $ and $\displaystyle \inf $ for those particular sets. What does $\displaystyle I_x $ represent? I know that its the largest interval, because the endpoints are the glb and lub.