Is every orientable surface a 2-folded covering of a non-orientable surface?

Question

We learned that every non-orientable surface can be covered by an orientable surface 2-foldedly. But what about the statement that "Is every orientable surface a 2-folded covering of a non-orientable surface?"

Answer 1

Recall that $k\mathbb{RP}^2$ is a non-orientable surface with Euler characteristic $2 - k$. Its orientable double cover is a closed orientable surface with Euler characteristic $2(2 - k) = 4 - 2k = 2 - 2(k - 1)$; by the classification of closed surfaces, it must be $\Sigma_{k-1}$. So every closed orientable surface occurs as the orientable double cover of some non-orientable surface.