Irrational Number Space is Non-Meager

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 non-meager.

Proof

From Irrational Number Space is Complete Metric Space, $\struct {\R \setminus \Q, d}$ is a complete metric space.

From the Baire Category Theorem, a complete metric space is also a Baire space.

The result follows from Baire Space is Non-Meager.

$\blacksquare$

Sources

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $31$. The Irrational Numbers: $6$