= Solution
A <Riemann surface> is a connected Hausdorff second-countable space equipped with complex coordinate charts whose changes of coordinate are holomorphic with holomorphic inverses. Such charts give it the structure of a one-dimensional <complex manifold>.
A holomorphic change of coordinate has nonzero derivative. In real coordinates its derivative matrix has determinant $|f'(z)|^2>0$ by the <Cauchy-Riemann equations>. Thus every overlap preserves the standard <orientation> of the <complex plane>. These chart orientations fit together, proving \b[every <Riemann surface> is <orientable>].
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