Is R2Q2\mathbb{R}^2 \setminus \mathbb{Q}^2 Homeomorphic To RQ\mathbb{R} \setminus \mathbb{Q}?

Let R2Q2\mathbb{R}^2 \setminus \mathbb{Q}^2 denote the Cartesian plane minus all points whose coordinates are both rational. Similarly, let RQ\mathbb{R} \setminus \mathbb{Q} denote the irrational numbers. Are the spaces R2Q2\mathbb{R}^2 \setminus \mathbb{Q}^2 and RQ\mathbb{R} \setminus \mathbb{Q} homeomorphic?


Hint: Is R2Q2\mathbb{R}^2 \setminus \mathbb{Q}^2 path-connected?

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