# Math Help - Open sets

1. ## Open sets

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

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

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

What the text is doing is constructing what are know as components in topology. A component is a maximal connected set. Components are pair-wise disjoint. In $\Re^1$ the components of an open subspace are open intervals. You are correct to worry about glb’s and lub’s. In the example you gave, unless $U$ is bounded, it is possible that $a_x = - \infty \mbox{ or } b_x = \infty$.