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:

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

$\blacksquare$

Also see

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

Sources

• 1975: Bert Mendelson: Introduction to Topology (3rd ed.) ... (previous) ... (next): Chapter $2$: Metric Spaces: $\S 6$: Open Sets and Closed Sets: Exercise $1$