Let . Since acts properly discontinuously, has an open neighbourhood such for any with . Then is an evenly covered open neighbourhood of : and is a homeomorphism for each , since it is surjective and continuous (by construction), injective (no to points in lie in the same orbit by proper continuity), and open (in fact is open, since open implies open for all , so is open and thus is open). Thus every has an evenly covered neighbourhood, so is a covering of the orbit space with itself.