1. analysis question

Show that every open set UR 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.]

can anyone help me to approach this question ? ,,,
thanks a lot

2. Originally Posted by jin_nzzang
Show that every open set UR 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 $\displaystyle x \in U$ define two sets $\displaystyle L_x = \left\{ {y:\left[ {x,y} \right) \subset U} \right\}\quad \& \quad K_x = \left\{ {y:\left( {y,x} \right] \subset U} \right\}$.
You be able to give reasons why both sets are nonempty.

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