Irrational Number Space is not Scattered

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 not scattered.

Proof
For a space to be scattered, it needs by definition to have no subset which is dense-in-itself.

From Irrational Number Space is Dense-in-itself, $\struct {\R \setminus \Q, \tau_d}$ is dense-in-itself.

Hence the result.