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 $\R$.

$\ds \bigcap_{\alpha \mathop \in \Q} B_\alpha^e$

$=$

$\ds \bigcap_{\alpha \mathop \in \Q} \R \setminus B_\alpha^-$

Definition 1 of Exterior

$\ds $

$=$

$\ds \bigcap_{\alpha \mathop \in \Q} \R \setminus B_\alpha$

$\ds $

$=$

$\ds \R \setminus \bigcup_{\alpha \mathop \in \Q} B_\alpha$

$\ds $

$=$

$\ds \R \setminus \Q$

$\blacksquare$