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

Then:
 * $\displaystyle \bigcup_{\alpha \mathop \in \Q} {B_\alpha}^- = \struct {\Q, \tau_d}$

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, $B_\alpha = \set \alpha$ is closed in $\R$.

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

Hence the result.