Union of Closures of Singleton Rationals is Rational Space

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

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

Then the union of the closures in the set of real numbers $\R$ of all $B_\alpha$ is $\Q$:
 * $\displaystyle \bigcup_{\alpha \mathop \in \Q} \map \cl {B_\alpha} = \Q$

Proof
Let $\alpha \in \Q$.

By Real Number is Closed in Real Number Space, $B_\alpha = \set \alpha$ is closed in $\R$.

From Closed Set Equals its Closure, it follows that:
 * $B_\alpha = \map \cl {B_\alpha}$

Hence the result.