For a nonsingular algebraic variety $X$ over the complex numbers, the functor $Y \mapsto Y^{an}$ which sends complex algebraic varieties to their complex analytic topology establishes an equivalence of categories between the corresponding étale sites $X_{et} \simeq X^{an}_{et}$.
James Milne, section 21 of Lectures on Étale Cohomology
Riemann’s existence theorem (pdf)
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