 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 $|f^{\prime }(z){|}^2>0$ by the Cauchy-Riemann equations. Thus every overlap preserves the standard orientation of the complex plane. These chart orientations fit together, proving every Riemann surface is orientable.