It exists a bijection $\psi:\{0,1\}^{\mathbb{N}}\to \mathbb{R}$. If on $\{0,1\}$ we put the discrete topology, then are the two spaces above homeomorphic considering the product topology on the first space?
2026-02-22 21:26:41.1771795601
$\{0,1\}^{\mathbb{N}}$ homeomorphic to $\mathbb{R}$?
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No, because, by Tychonoff's theorem, $\{0,1\}^{\mathbb N}$ is compact.