Is \(\mathbb{R}^2 \setminus \mathbb{Q}^2\) Homeomorphic To \(\mathbb{R} \setminus \mathbb{Q}\)?

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

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

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