I am trying to prove that $\{0,1\}^\mathbb{N}$ is homeomorphic to the cantor set. Consider the mapping $f:\{0,1\}^\mathbb{N}\to[0,1]$ defined as$$f(x)=2\sum_{n=0}^\infty 3^{-n-1}x(n)$$ I think that $f$ being continuous and injective would be enough. how does one show it is injective?
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Indeed, I found my answer there! – user91083 Aug 20 '13 at 09:37