is a principal , where is considered as a discrete topological group. Every universal cover is regular, with deck transformation group being isomorphic to the fundamental group Examples Let be the covering for some , then the map is a deck transformation and . Let be the covering , then the map with is a deck transformation and . As another important example, consider the complex plane and the complex plane minus the origin. Then the map with is a regular cover. The deck transformations are multiplications with -th roots of unity and the deck transformation group is therefore isomorphic to the cyclic group . Likewise, the map with is the universal cover. Properties Let be a path-connected space and be a connected covering. Since a deck transformation is bijective, it permutes the elements of a fiber with and is uniquely determined by where it sends a single point. In particular, only the identity map fixes a point in the fiber. Because of this property every deck transformation defines a group action on , i.e. let be an open neighborhood of a and an open neighborhood of an , then is a group action. Normal coverings Definition A covering is called normal, if . This means, that for every and any two there exists a deck transformation , such that . Properties Let be a path-connected space and be a connected covering. Let be a subgroup of , then is a normal covering iff is a normal subgroup of . If is a normal covering and , then . If is a path-connected covering and , then , whereby is the normaliser of . Let be a topological space. A group acts discontinuously on , if every has an open neighborhood with , such that for every with one has . If a group acts discontinuously on a topological space , then the quotient map with is a normal covering. Hereby is the quotient space and is the orbit of the group action. Examples The covering with is a normal coverings for every . Every simply connected covering is a normal covering. Calculation Let be a group, which acts discontinuously on a topological space and let be the normal covering. If is path-connected, then . If is simply connected, then . Examples Let . The antipodal map with generates, together with the composition of maps, a group and induces a