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I've been given the definition that $X$ and $Y$ are homomorphic if there is a bijection between them, with both the function and inverse being continuous.

Haven't really looked at them other than that, was just given this for some work. I don't need to worry about showing that the functions are continuous though, just come up with homorphism for two spaces. Here's some of them:

$1$) $S^1 \times \mathbb{R}$ and the subspace $\{z \in \mathbb{C}: z \neq 0\}$ of $\mathbb{C}$.

$2$) The torus $S^1 \times S^1$ and the subspace $\{z_1 \times z_2 \in \mathbb{C} \times \mathbb{C}:z_1 \bar{z_1} = 1, z_2 \bar{z_2} = 1\}$ of $\mathbb{C} \times \mathbb{C}$

I'm not really sure how I should go about thinking of these questions to define a function that does what we want.

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Homeomorphic, not homomorphic. –  Zev Chonoles Sep 20 '13 at 4:26
    
Question (2) doesn't look right. Are you sure you copied it down correctly? –  Chris Culter Sep 20 '13 at 4:29
    
Yes, you're right. That was from another question, I've fixed it now. –  MangoPirate Sep 20 '13 at 4:35
    
Hint : For $z\in \mathbb{C}$, write $z =re^{i\theta}$ –  Prahlad Vaidyanathan Sep 20 '13 at 5:09

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