  We show that every orientable infinite-type surface is properly rigid as a
consequence of a more general result. Namely, we prove that if a homotopy
equivalence between any two non-compact orientable surfaces is a proper map,
then it is properly homotopic to a homeomorphism, provided surfaces are neither
the plane nor the punctured plane. Thus all non-compact orientable surfaces,
except the plane and the punctured plane, are topologically rigid in a strong
sense.
