Exterior of Union of Singleton Rationals is Empty

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

Let $\left({\R, \tau_d}\right)$ be the real number line under the [Definition:Euclidean Topology on Real Number Line|usual (Euclidean) topology]] $\tau_d$.

Then the exterior in $\left({\R, \tau_d}\right)$ of the union of all $B_\alpha$ is the empty set:
 * $\displaystyle \left({\bigcup_{\alpha \mathop \in \Q} B_\alpha}\right)^e = \varnothing$

Proof
By definition:
 * $B_\alpha = \left\{{\alpha}\right\}$

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

By definition, the exterior of $\Q$ is the complement of the closure of $\Q$ in $\R$.

By Closure of Rational Numbers is Real Numbers:
 * $\Q^- = \R$

By Relative Complement with Self is Empty Set:
 * $\complement_\R \left({\R}\right) = \varnothing$

Hence the result.