# Proof that every open set in $\mathbb{R}^n$ is the union of …??

Can't figure this 2 part question out - I think the first part involves using open balls but I'm not sure. It's straightforward to prove this for $\mathbb{R}^1$ with disjoint segments but am a little lost as to what to do when it comes to $\mathbb{R}^n$.

(1) Prove that every open set in $\mathbb{R}^n$ is the union of an at most countable collection of disjoint segments. Hint: You need to replace disjoint segments with the appropriate objects (which I'm thinking are open balls).

(2) Generalize the statement to a separable topological space. Hint: you may need some extra assumption.

Any help would be much appreciated! The more detail, the better! Thanks!

-
What exactly is a segment? –  arkeet Dec 14 '12 at 7:12

Disjoint?${}{}$ –  tomasz Feb 5 at 22:45