Intersection of Exteriors of Singleton Rationals is Irrationals

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:
 * $\ds \bigcap_{\alpha \mathop \in \Q} B_\alpha^e = \R \setminus \Q$

where $B_\alpha^e$ denotes the exterior of $B_\alpha$ in $\R$.