Closure of Union of Singleton Rationals is Real Number Line

Theorem
Let $B_\alpha$ be the singleton containing the rational number $\alpha$.

Then the closure of the union of all $B_\alpha$ is the set of real numbers:
 * $\displaystyle \left({\bigcup_{\alpha \mathop \in \Q} B_\alpha}\right)^- = \R$

Proof
By definition:
 * $B_\alpha = \left\{{\alpha}\right\}$

Thus:
 * $\displaystyle \bigcup_{\alpha \mathop \in \Q} B_\alpha = \Q$

The result follows from Closure of Rational Numbers is Real Numbers.