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$\begingroup$

I'd like a hint to prove that this function is a homeomorphism: $$f[z:w]=\left(\frac{\operatorname{Re}( w \bar{z})}{|w|^2 + |z|^2}, \frac{\operatorname{Im}(w\bar{z})}{|w|^2 + |z|^2},\frac{|w|^2-|z|^2}{|w|^2+|z|^2}\right)$$ of $\mathbb{P}^1$ onto $\mathbb{S}^2$. Thanks.

ADDED(06/27/12): The previous definition of $f$ was wrong, this new one seems to work...

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    Hint: prove that it's continuous, one-one, onto, and has a continuous inverse. In case you already knew that, which of those four parts can you do, and which one(s) give you trouble?2012-06-27
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    If you multiply each of $w$ and $z$ by $100$, you still have the same point in $\mathbb{P}^1$, and the first two components of this triple become $100^2$ times as big, but the third one stays the same, so this can't be right.2012-06-27
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    @MichaelHardy this is indeed true, this is Problem I.2 C from Miranda's book on Riemann surfaces and algebraic curves, as you said, something isn't right...2012-06-27
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    @GerryMyerson I don't know how to prove the continuity, of $f$ and its inverse(I'm still trying to find its inverse)2012-06-27
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    I see you've now put in the denominators in the first two components, so maybe now it works.2012-06-27
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    So far I'm the only person who's up-voted this question.2012-06-27

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