Handout 1
... about the Klein bottle as two Möbius bands glued together along their boundary circles. (6) Let X = I 2 be the unit square with the equivalence relation, (t, 0) ∼ (1 − t, 1) and (0, t) ∼ (1, 1 − t) for all 0 6 t 6 1, gluing the opposite sides in pairs and reversing the orientation of both pairs. Th ...
... about the Klein bottle as two Möbius bands glued together along their boundary circles. (6) Let X = I 2 be the unit square with the equivalence relation, (t, 0) ∼ (1 − t, 1) and (0, t) ∼ (1, 1 − t) for all 0 6 t 6 1, gluing the opposite sides in pairs and reversing the orientation of both pairs. Th ...
Theorem 2.1. Tv is a topologyfor v. Definition. For each x EX, 1T(X)is
... mutually exclusive subsets of X whose union is X (i.e. a partition of X). We wish to define a new topological space whose points are the members of V. To that end we define a topology Tv on V as follows: a subset F of V belongs to Tv if and only if F* belongs to T. ...
... mutually exclusive subsets of X whose union is X (i.e. a partition of X). We wish to define a new topological space whose points are the members of V. To that end we define a topology Tv on V as follows: a subset F of V belongs to Tv if and only if F* belongs to T. ...
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.