Question
Show that every open subset of a metric space can be expressed as a union of open
balls.
So far I have the following:
"Let $U \subseteq X$. For each $a \in U$, choose $r_a > 0$ such that $B(a, r_a) \subseteq U$."
I'm just not sure what the next step to show that $\bigcup_{a \in U}B(a, r_a) = U$.
Best Answer
$\bigcup_{a \in U}B(a, r_a) \subseteq U$, because it is, by definition, an union of subsets $B(a, r_a)$ of $U$.
$U \subseteq \bigcup_{a \in U}B(a, r_a)$, because each $a \in U$ is also contained in a right-hand side.