Open subsets are a union of disjoint open intervals

[redone632]

Homework Statement

Prove that any open subset of $\Real$ can be written as an at most countable union of disjoint open intervals.

Homework Equations

An at most countable set is either finite or infinitely countable.

The Attempt at a Solution

It seems very intuitive but I am at lost where to even start. We're doing compactness in metric spaces so I would assume it must apply. But I thought a set has to be closed in order to be compact and this deals with an open subset so it can't possibly be compact. Any help would be much appreciated!

 

[Office_Shredder]

This has pretty much nothing to do with compactness. Try this:
1) Show you can write it as a union of disjoint intervals.
2) Think about rational numbers

 

[redone632]

Hmmm, I think I've made some development but I'm still unsure how solid it is.

Let $E$ be an open subset of $\Real$. Define the equivalence relation $x \sim y \Longleftrightarrow \exists (a,b)$ such that ${x,y} \in (a,b) \subseteq E$. The equivalence classes will be distinct. By the archimedean property, there exists a distinct rational number in each interval. Since the rationals are countable, we have a countable number of intervals. The countable union of these intervals will equal the set E. Therefore, every subset of R can be represented by a countable union of disjoint open intervals.