# Is the Coordinate Chart Always a Biholomorphism?

Let $X$ be a Riemann surface with complex structure $\{(U_i,\phi_i)\}$. Is it the case that $\phi_i:U_i\rightarrow V_i$ is a biholomorphic map in the sense of Riemann surfaces?

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The answer is obviously yes. Indeed we only need check that $\phi_i^{-1}\phi_i:V_i\rightarrow V_i$ is holomorphic in the classical sense, which it trivially is.