Lecture 2. Smooth functions and maps
... each circle through D1 by an angle θ1 in such a way that the intersection point with D1 has θ1 = 0, and similarly for D2 . Points on the northern hemisphere of S 2 correspond to those circles which pass through a smaller disk D̃1 in D1 , and those on the southern hemisphere correspond to circles pas ...
... each circle through D1 by an angle θ1 in such a way that the intersection point with D1 has θ1 = 0, and similarly for D2 . Points on the northern hemisphere of S 2 correspond to those circles which pass through a smaller disk D̃1 in D1 , and those on the southern hemisphere correspond to circles pas ...
Covering space
In mathematics, more specifically algebraic topology, a covering map (also covering projection) is a continuous function p from a topological space, C, to a topological space, X, such that each point in X has an open neighbourhood evenly covered by p (as shown in the image); the precise definition is given below. In this case, C is called a covering space and X the base space of the covering projection. The definition implies that every covering map is a local homeomorphism.Covering spaces play an important role in homotopy theory, harmonic analysis, Riemannian geometry and differential topology. In Riemannian geometry for example, ramification is a generalization of the notion of covering maps. Covering spaces are also deeply intertwined with the study of homotopy groups and, in particular, the fundamental group. An important application comes from the result that, if X is a ""sufficiently good"" topological space, there is a bijection between the collection of all isomorphism classes of connected coverings of X and the conjugacy classes of subgroups of the fundamental group of X.