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Math Help - Homeomorphisms..

  1. #1
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    Homeomorphisms..

    Problem:

    Construct a homeomorphism \phi : \{0,1\}^{\mathbb{N}} \to \{0,1,2\}^{\mathbb{N}} with the product topology on both spaces.

    Tried thinking of a few simple examples so far, but none were satisfactory. Any clues\hints?

    Thanks, and happy holidays.
    Last edited by Defunkt; December 28th 2009 at 01:33 PM.
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  2. #2
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    Quote Originally Posted by Defunkt View Post
    Problem:

    Construct a homeomorphism \phi : \{0,1\}^{\mathbb{N}} \to \{0,1,2\}^{\mathbb{N}} with the product topology on both spaces.
    I don't see much hope of getting a simple formula for such a map. The obvious inclusion map from {0,1} to {0,1,2} gives an injective map f:\{0,1\}^{\mathbb{N}} \to \{0,1,2\}^{\mathbb{N}}. Also, there is an obvious injective map from {0,1,2} to {0,1}×{0,1}, and this extends to an injective map g:\{0,1,2\}^{\mathbb{N}} \to \bigl(\{0,1\}\times\{0,1\}\bigr)^{\mathbb{N}} \cong  \{0,1\}^{\mathbb{N}}. You could try applying one of the constructive proofs of the Cantor–Schröder–Bernstein theorem to this pair of maps in order to find a bijective map (which would probably turn out to be a homeomorphism). I haven't tried to do that, and I don't know whether it would be workable, but I can't think of any other approach to the problem.
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  3. #3
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    Hi Opalg, sorry for late response but thanks!

    We ended up with \Phi : \{0,1\}^{\mathbb{N}} \to \{0,1,2\}^{\mathbb{N}},  \Phi \left( \{ a_n \}_{n=1}^{\infty} \right) =  \{ \phi_n \}_{n=1}^{\infty} \, \text{ where } \ \phi_i = \displaystyle \begin{cases} 0 & a_i=0 \\ 1 & a_i = 1 \mbox{ and } a_{i+1}=0 \\ 2 & a_i = a_{i+1}=1 \end{cases}

    Which apparently is a homeomorphism on the product topology
    Last edited by Defunkt; January 5th 2010 at 08:55 AM.
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