Union of Open Sets of Metric Space is Open

Theorem
Let $M = \left({A, d}\right)$ be a metric space.

The union of a set of open sets of $M$ is open in $M$.

Proof
Let $I$ be any indexing set.

Let $U_i$ be open in $M$ for all $i \in I$.

Let $\displaystyle x \in \bigcup_{i \mathop \in I} U_i$.

Then $x \in U_k$ for some $k \in I$.

Since $U_k$ is open in $M$:
 * $\displaystyle \exists \epsilon > 0: B_\epsilon \left({x}\right) \subseteq U_k \subseteq \bigcup_{i \mathop \in I} U_i$

where $B_\epsilon \left({x}\right)$ is the open $\epsilon$-ball of $x$ in $M$.

The result follows.