Closure of Union of Singleton Rationals is Real Number Line

Theorem
Let $\struct {\Q, \tau_d}$ be the rational number space under the usual (Euclidean) topology $\tau_d$.

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

Then the closure in the set of real numbers $\R$ of the union of all $B_\alpha$ is $\R$ itself:


 * $\ds \map \cl {\bigcup_{\alpha \mathop \in \Q} B_\alpha} = \R$

Proof
By definition:
 * $B_\alpha = \set \alpha$

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

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