# 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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