A simple bijection between $\mathbb{R}$ and $\mathbb{R}^4$ or $\mathbb{R}^n$?

cardinals

Let $f$ be any bijection from $\mathbb{R}$ to $P(\mathbb{N})$ (the simplest one I could think of uses continued fractions, e.g. see here). To construct a bijection from $\mathbb{R}$ to $\mathbb{R}^n$ take $$g_i(A) = \left\{ \left.\frac{x-i}{n} \right|\ x\in A, x =i\ (\mathrm{mod}\ n) \right\}$$ and set $$h_i = f^{-1} \circ g_i \circ f$$ and then $$h(x) = \langle h_0(x), h_1(x), \ldots, h_{n-1}(x) \rangle$$ will be your bijection.

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