Open set as a countable union of spheres Could you tell me how to prove that each open set can  is a countable union of open balls: $ K\left( x,r \right) = \left\{ y  \in \mathbb{R} ^{n} | \ d(x,y) <r \right\}$ where d is the Euclid metric?
I know that  $U$ is an open set if $\ int \ U=U$ 
and for $E \subset \mathbb{R}, \ \ $ $intE=\bigcup \left\{K \ | \ K\subset E \ \right\} $ (K as defined above).

Prove that $U\cap \mathbb Q^n$ is dense in $U$. Then consider the family $\mathcal A =\left\{K(z,q):z\in U\cap \mathbb Q^n, q \in \mathbb Q,K(z,q)\subseteq U\right\}$.
Prove that $\mathcal A $ is countable and $\cup \mathcal A=U.$ 
