analysis question

Quote:

Originally Posted by jin_nzzang

Show that every open set U⊆R is a union of at most countably many disjoint open intervals [Hint: Each point x of U lies in a unique maximal interval of U, which is the union of all intervals contained in U which contain x. Each such interval contains at least one rational number.]

This problem is completely solved by using the hint.
The interesting part of this is proving the hint is true.
If $x \in U$ define two sets $L_x = \left\{ {y:\left[ {x,y} \right) \subset U} \right\}\quad \& \quad K_x = \left\{ {y:\left( {y,x} \right] \subset U} \right\}$ .
You be able to give reasons why both sets are nonempty.

Now look for least upper bound of $L_x$ and greatest lower bound of $K_x$ and with those as endpoints you have a maximal open interval containing $x$ and is a subset of $U$ .