holomorphic map and is a connected and simply connected Riemann surface. Factorisation Let and be path-connected, locally path-connected spaces, and and be continuous maps, such that the diagram commutes. If and are coverings, so is . If and are coverings, so is . Product of coverings Let and be topological spaces and and be coverings, then with is a covering. Equivalence of coverings Let be a topological space and and be coverings. Both coverings are called equivalent, if there exists a homeomorphism , such that the diagram commutes. If such a homeomorphism exists, then one calls the covering spaces and isomorphic. Lifting property An important property of the covering is, that it satisfies the lifting property, i.e.: Let be the unit interval and be a covering. Let be a continuous map and be a lift of , i.e. a continuous map such that . Then there is a uniquely determined, continuous map for which and which is a lift of , i.e. . If is a path-connected space, then for it follows that the map is a lift of a path in and for it is a lift of a homotopy of paths in . Because of that property one can show, that the fundamental group of the unit circle is an infinite cyclic group, which is generated by the homotopy classes of the loop with . Let be a path-connected space and be a connected covering. Let be any two points, which are connected by a path , i.e. and . Let be the unique lift of , then the map with is bijective. If is a path-connected space and a connected covering, then the induced group homomorphism with , is injective and the subgroup of consists of the homotopy classes of loops in , whose lifts are loops in . Branched covering Definitions Holomorphic maps between Riemann surfaces Let and be Riemann surfaces, i.e. one dimensional complex manifolds, and let be a continuous map. is holomorphic in a point , if for any charts of and of , with , the map is holomorphic. If is holomorphic at all , we say is holomorphic. The map is called the local expression of in . If is a non-constant, holomorphic map between compact Riemann surfaces, then is surjective and an open map, i.e. for every open set the image is also open. Ramification point and branch point Let be