Closure of Infinite Union may not equal Union of Closures/Proof 2

Proof
Proof by Counterexample:

Let $\struct {\R, \tau_d}$ denote the real number line under the usual (Euclidean) topology $\tau_d$.

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

For a rational number $\alpha \in \Q$, let $B_\alpha$ denote the singleton containing $\alpha$.

From Union of Closures of Singleton Rationals is Rational Space:


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

From Closure of Union of Singleton Rationals is Real Space:


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

So it is seen that:
 * $\displaystyle \bigcup_{\alpha \mathop \in \Q} \map \cl {B_\alpha} \subset \displaystyle \map \cl {\bigcup_{\alpha \mathop \in \Q} B_\alpha}$

but it is not the case that:
 * $\displaystyle \bigcup_{\alpha \mathop \in \Q} \map \cl {B_\alpha} = \displaystyle \map \cl {\bigcup_{\alpha \mathop \in \Q} B_\alpha}$

and the result is apparent.