1. ## Homeomorphisms..

Problem:

Construct a homeomorphism $\displaystyle \phi : \{0,1\}^{\mathbb{N}} \to \{0,1,2\}^{\mathbb{N}}$ 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.

2. Originally Posted by Defunkt
Problem:

Construct a homeomorphism $\displaystyle \phi : \{0,1\}^{\mathbb{N}} \to \{0,1,2\}^{\mathbb{N}}$ 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 $\displaystyle f:\{0,1\}^{\mathbb{N}} \to \{0,1,2\}^{\mathbb{N}}$. Also, there is an obvious injective map from {0,1,2} to {0,1}×{0,1}, and this extends to an injective map $\displaystyle g:\{0,1,2\}^{\mathbb{N}} \to \bigl(\{0,1\}\times\{0,1\}\bigr)^{\mathbb{N}} \cong \{0,1\}^{\mathbb{N}}$. 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.

3. Hi Opalg, sorry for late response but thanks!

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

Which apparently is a homeomorphism on the product topology