Open Balls of Same Radius form Open Cover

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

Let $\UU_\epsilon = \set{\map {B_\epsilon} x : x \in A}$

That is, $\UU_\epsilon$ is the set of all open balls of radius $\epsilon > 0$.

Then:
 * $\UU_\epsilon$ is an open cover of $M$.

Proof
From Open Ball is Open Set:
 * $\UU_\epsilon$ is a set of open subsets

From Center is Element of Open Ball:
 * $\forall x \in A : x \in \map {B_\epsilon} x$

By definition, $\UU_\epsilon$ is a cover of $A$.

By definition, $\UU_\epsilon$ is an open cover of $A$.