Q:

Decomposition of an open set

Let A be an open set.  Can A always be described as the union of (possibly infinite) balls? 

A:

In metric space, for each $x\in A$, there is a ball $B(x,r_{x})$ such that $x\in B(x,r_{x})\subseteq A$.  Having identified $x\rightarrow B(x,r_{x})$, then it is ready to see that $A=\displaystyle\bigcup_{x\in A}B(x,r_{x})$.

