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Exercises
Exercises

Week 5 Term 2
Week 5 Term 2

... Proof. First we remark that deck transformations of a covering space obviously have the properly discontinuous property. To prove the result, take any open set U as in the definition of proper discontinuity. Then the quotient map identifies the disjoint homeomorphic neighbourhoods {g(U ) : g ∈ G} wi ...
PDF
PDF

Abstract. An A-spectral space is a space such that its... compactification is a spectral space. An up-spectral space is de-
Abstract. An A-spectral space is a space such that its... compactification is a spectral space. An up-spectral space is de-

... ...
June 2012
June 2012

The final exam
The final exam

... (Write the answer as a free product of two well-known groups.) (f) The wireframe ...
1. (a) Let X be a topological space and γ 0,γ1 : [0
1. (a) Let X be a topological space and γ 0,γ1 : [0

Topology I – Problem Set Five Fall 2011
Topology I – Problem Set Five Fall 2011

Math 571 Qualifying Exam 1. Let (Y,T ) be a topological space, and
Math 571 Qualifying Exam 1. Let (Y,T ) be a topological space, and

. TOPOLOGY QUALIFYING EXAMINATION Time: Three hours.
. TOPOLOGY QUALIFYING EXAMINATION Time: Three hours.

... (a) Compute the fundamental group of the space obtained by identifying the points x and y pictured above on the solid three hole torus. (b) Compute the fundamental group of the space obtained by identifying the points x and y on the surface of the three hole torus. 2. (a) Define the join (or product ...
HOMEWORK 7 Problem 1: Let X be an arbitrary nonempty set
HOMEWORK 7 Problem 1: Let X be an arbitrary nonempty set

... Problem 2: Let U and V be two path-connected open subsets of Rn such that U ∪ V = Rn . Show that U ∩ V is path-connected. Solution: This is a straight-up Mayer-Vietoris question. We have a long exact sequence · · · → H1 (Rn ) → H0 (U ∩ V ) → H0 (U ) ⊕ H0 (V ) → H0 (Rn ) → 0. Now H1 (Rn ) = 0 since R ...
Products and quotients via universal property
Products and quotients via universal property

PDF
PDF

QUALIFYING EXAM IN TOPOLOGY WINTER 1996
QUALIFYING EXAM IN TOPOLOGY WINTER 1996

... b) Is it true that f∗ : H1 (X) → H1 (Y ) is a monomorphism? Give either a proof or a counterexample. 3. Let X denote the set of all real numbers with the finite-complement topology, and define f : E 1 → X by f (x) = x. Show that f is continuous, but not a homeomorphism. 4. Given a continuous map f : ...
G-sets, G-spaces and Covering Spaces
G-sets, G-spaces and Covering Spaces

Click here
Click here

... 3. DISprove the following assertion: Let τ and τ 0 be two topologies on the same space X. Then if (X, τ ) ≈ (X, τ 0 ), then τ = τ 0 . 4. A discrete map is a continuous function d : X → D, where D is a finite set given the discrete topology. Prove that X is connected if and only if every discrete map ...
UNIFORMIZATION OF SURFACES COMPLEX ANALYSIS 8702 1. Riemann surfaces; Summary
UNIFORMIZATION OF SURFACES COMPLEX ANALYSIS 8702 1. Riemann surfaces; Summary

June 2010
June 2010

... 1. Let X be a topological space and let f, g : X → R be continuous functions. (a): Show that the set L = {p ∈ X : f (p) ≤ g(p)} is a closed subset of X. (b): Show that the function h : X → R given by h(p) = min{f (p), g(p)} is continuous. 2. Let X be a Hausdorff space, p ∈ X, and A ⊆ X a compact sub ...
Let X be a path-connected space and suppose that every map f: S^1
Let X be a path-connected space and suppose that every map f: S^1

Qualifying Exam in Topology
Qualifying Exam in Topology

... 3. (a) Define the two notions: “homotopy between two maps” and “homotopy equivalence between two topological spaces.” (b) Give an example of topological spaces X and Y that have the same homotopy type but are not homeomorphic. (c) Give an example of (path-connected) topological spaces X and Y that h ...
abs
abs

Qualifying Exam in Topology January 2006
Qualifying Exam in Topology January 2006

TOPOLOGY QUALIFYING EXAM carefully.
TOPOLOGY QUALIFYING EXAM carefully.

PDF
PDF

Math 8301, Manifolds and Topology Homework 7
Math 8301, Manifolds and Topology Homework 7

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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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