Union of Closures of Singleton Rationals is Rational Space

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

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

Then:
 * $\displaystyle \bigcup_{\alpha \mathop \in \Q} B_\alpha^- = \left({\Q, \tau_d}\right)$

where $B_\alpha^-$ denotes the closure of $B_\alpha$ in $\R$.

Proof
Let $\alpha \in \Q$.

By Real Number is Closed in Real Number Space, $B_\alpha = \left\{{\alpha}\right\}$ is closed in $\R$.

From Closed Set Equals its Closure, it follows that:
 * $B_\alpha = B_\alpha^-$

Hence the result.