Irrational Number Space is Separable

Theorem
Let $\struct {\R \setminus \Q, \tau_d}$ be the irrational number space under the Euclidean topology $\tau_d$.

Then $\struct {\R \setminus \Q, \tau_d}$ is separable.

Proof
Let $S$ be the set defined as:
 * $S = \set {\pi + q: q \in \Q}$

From Rational Numbers are Countably Infinite, $\Q$ is countable.

Therefore $S$ is also countable.

From $\pi$ is Irrational:
 * $\pi \in \R \setminus \Q$

It follows from Rational Addition is Closed that:
 * $\forall q \in \Q: \pi + q \in \R \setminus \Q$

and so:
 * $S \subseteq \R \setminus \Q$

From Rationals plus Irrational are Everywhere Dense in Irrationals, it follows that $S$ is everywhere dense in $\R \setminus \Q$.

Thus we have constructed a countable subset $S$ of $\R \setminus \Q$ which is everywhere dense in $\R \setminus \Q$.

The result follows by definition of separable space.