The Orientation Double Covering Space

We describe the orientation double covering space of a manifold from differ-ent angles and show why they are equivalent.Moreover,we will explain how to construct a general 2-fold covering space of a manifold using cut-and-paste strat-egy.In the last section,We will describe the orientation double covering space by the Cech cohomology and give a short remark about the relationship between the orientation covering double space and the first Stiefel-Whitney class.