Prove that $$ f: \prod\limits_{1}^{\infty} ( \{0,2 \}, \mathcal{T} _{\delta}) \to ([0,1], \mathcal{T}_{e}):\{n_i \} \mapsto \sum_{i=1}^{\infty} \frac{n_i}{3^i} $$ is homeomorphism, and image of $f$ is Cantor set.
Prove that function is homeomorphism.
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general-topology
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0If you need some more help, try posting a comment to Bryan's answer asking for it, or edit your post to ask specifically for help with the continuity of $f$ and $f^{-1}$. In particular, you *shouldn't* post an "answer" of the sort you have below (Bryan won't be notified it's there, so it doesn't help you), nor should you repost basically the same question slightly narrowed. – 2012-11-04
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0Ok. Sorry for that. I'm new here. Now I will respect this rules. – 2012-11-04
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0It's quite all right--I've done similar things "against protocol" before I knew better. Just wanted to fill you in. – 2012-11-04
1 Answers
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Since you’ve given no indication of where you’re having trouble, I’ll give a couple of pointers to previous answers that should at least get you started. Here is a proof that the map is injective. This answer contains most of what you need to prove that the range of $f$ is the familiar middle-thirds Cantor set.
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0Thank you Brian M. Scott :). Now I know how to prove that range of f is Cantor set and that this function is injective, but I still don't know how to prove that $f, f^{-1}$ are continuous. Can you help me one more time? – 2012-11-04