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18.906 Problem Set 4 Alternate Question
18.906 Problem Set 4 Alternate Question

Jerzy DYDAK Covering maps for locally path
Jerzy DYDAK Covering maps for locally path

... Abstract. We define Peano covering maps and prove basic properties analogous to classical covers. Their domain is always locally pathconnected but the range may be an arbitrary topological space. One of characterizations of Peano covering maps is via the uniqueness of homotopy lifting property for a ...
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... The universal covering space has the following universal property: If π : (X̃, x0 ) → (X, x) is a based universal cover, then for any connected based cover π 0 : (X 0 , x0 ) → (X, x), there is a unique covering map π 00 : (X̃, x0 ) → (X 0 , x0 ) such that π = π 0 ◦ π 00 . Clearly, if a universal cov ...
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PDF

Let X and Y be topological spaces, where the only open
Let X and Y be topological spaces, where the only open

Math 8301, Manifolds and Topology Homework 8 1. Show that S
Math 8301, Manifolds and Topology Homework 8 1. Show that S

... 1. Show that S 2 is isomorphic to the universal covering space of RP2 . 2. Give a description of the universal cover of the space S 2 ∨ S 1 , obtained by gluing together S 2 and S 1 at a single point. 3. Suppose X and Y are path-connected spaces, p : Y → X is a covering map, and y ∈ Y . Let the imag ...
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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.
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