Decomposition of an open set

Question

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

Answer 1

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 $A=\displaystyle\bigcup_{x\in A}B(x,r_{x})$.

Answer 2

In the Euclidean topology yes, because open balls are a basis for the topology. The definition you should look for is the definition of basis for a topology.

More generally, in any metric space, the open balls form a basis for the topology. But in general topological spaces, even the concept of open balls may not make sense.