group action , which acts discontinuously on . Because of it follows, that the quotient map is a normal covering and for a universal covering, hence for . Let be the special orthogonal group, then the map is a normal covering and because of , it is the universal covering, hence . With the group action of on , whereby is the semidirect product , one gets the universal covering of the klein bottle , hence . Let be the torus which is embedded in the . Then one gets a homeomorphism , which induces a discontinuous group action , whereby . It follows, that the map is a normal covering of the klein bottle, hence . Let be embedded in the . Since the group action is discontinuously, whereby are coprime, the map is the universal covering of the lens space , hence . Galois correspondence Let be a connected and locally simply connected space, then for every subgroup there exists a path-connected covering with . Let and be two path-connected coverings, then they are equivalent iff the subgroups and are conjugate to each other. Let be a connected and locally simply connected space, then, up to equivalence between coverings, there is a bijection: For a sequence of subgroups one gets a sequence of coverings . For a subgroup with index , the covering has degree . Classification Definitions Category of coverings Let be a topological space. The objects of the category are the coverings of and the morphisms between two coverings and are continuous maps , such that the diagram commutes. G-Set Let be a topological group. The category is the category of sets which are G-sets. The morphisms are G-maps between G-sets. They satisfy the condition for every . Equivalence Let be a connected and locally simply connected space, and be the fundamental group of . Since defines, by lifting of paths and evaluating at the endpoint of the lift, a group action on the fiber of a covering, the functor is an equivalence of categories. Applications An important practical application of covering spaces occurs in charts on SO(3), the rotation group. This group occurs widely in engineering, due to 3-dimensional rotations being heavily used in navigation, nautical engineering, and aerospace engineering, among many other uses. Topologically, SO(3) is the real projective space RP3, with fundamental group Z/2, and only (non-trivial) covering space the hypersphere S3, which