Open Balls form Basis for Open Sets of Metric Space

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

Let $\mathcal B$ be the set of all open balls of $M$.

Then $\mathcal B$ is a basis for the open sets of $M$.

Proof
Let $U$ be an open set of $M$.

Then by definition:
 * $\forall y \in U: \exists \epsilon \in \R_{>0}: B_\epsilon \left({y}\right) \subseteq U$

Thus:
 * $\displaystyle U = \bigcup_{y \mathop \in U} B_\epsilon \left({y}\right)$