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How do one show that Riemann sphere $=\mathbb{C} \cup\{\infty\}$ has a unique structure of Riemann surface?

To show $\mathbb{C} \cup\{\infty\}$ is a Riemann surface I just need to set up two charts, they are $\{U_1,f_1\}$ and $\{U_2,f_2\}$ with

$$U_1=\mathbb{C},~~~~f_1:U_1 \to \mathbb{C},~~~~f_1(z)=z$$

$$U_2=\mathbb{C \setminus \{0\}\cup\{\infty\}},~~~~f_1:U_2 \to \mathbb{C},~~~~f_1(z)=\frac{1}{z}$$

And the transition maps $f_1\circ f_2^{-1}, f_2\circ f_1^{-1}$ from $\mathbb{C}\setminus\{0\}$ to $\mathbb{C}\setminus\{0\}$ with rule $z\mapsto {1\over z}$ are holomorphic. How about the unique part? And what does the word "structure" means? Can I read the question this way: Show that the Riemann sphere is a unique Riemann surface?

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  • $\begingroup$ The Riemann has only one complex structure, this is a consequence of Riemann-Roch theorem. $\endgroup$ – jimbo Sep 7 '15 at 14:49

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