= Cantor space
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{title2=$\{0,1\}^{\mathbb N}$}
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= Bernoulli space
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Cantor space is the binary sequence space with the <product topology> of discrete two-point factors. Here the Bernoulli-space synonym refers to this <topology>, without selecting a particular <probability measure>. The <metric> $d(x,y)=\sum_{j\ge1}2^{-j}|x_j-y_j|$ induces the <topology>. A diagonal subsequence argument gives <compactness>. Fixing a finite prefix gives a <clopen> <cylinder set>. The map $x\mapsto\sum_j2x_j3^{-j}$ is a homeomorphism to the usual <Cantor set>.
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