Open Balls form Basis for Open Sets of Metric Space

Theorem
Let $M = \struct {A, d}$ be a metric space.

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

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

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

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

Thus:
 * $\ds U = \bigcup_{y \mathop \in U} \map {B_\epsilon} y$

Also see

 * Open Balls on Rational Centers form Basis for Usual Topology on Plane