• Study Resource
  • Explore
    • Arts & Humanities
    • Business
    • Engineering & Technology
    • Foreign Language
    • History
    • Math
    • Science
    • Social Science

    Top subcategories

    • Advanced Math
    • Algebra
    • Basic Math
    • Calculus
    • Geometry
    • Linear Algebra
    • Pre-Algebra
    • Pre-Calculus
    • Statistics And Probability
    • Trigonometry
    • other →

    Top subcategories

    • Astronomy
    • Astrophysics
    • Biology
    • Chemistry
    • Earth Science
    • Environmental Science
    • Health Science
    • Physics
    • other →

    Top subcategories

    • Anthropology
    • Law
    • Political Science
    • Psychology
    • Sociology
    • other →

    Top subcategories

    • Accounting
    • Economics
    • Finance
    • Management
    • other →

    Top subcategories

    • Aerospace Engineering
    • Bioengineering
    • Chemical Engineering
    • Civil Engineering
    • Computer Science
    • Electrical Engineering
    • Industrial Engineering
    • Mechanical Engineering
    • Web Design
    • other →

    Top subcategories

    • Architecture
    • Communications
    • English
    • Gender Studies
    • Music
    • Performing Arts
    • Philosophy
    • Religious Studies
    • Writing
    • other →

    Top subcategories

    • Ancient History
    • European History
    • US History
    • World History
    • other →

    Top subcategories

    • Croatian
    • Czech
    • Finnish
    • Greek
    • Hindi
    • Japanese
    • Korean
    • Persian
    • Swedish
    • Turkish
    • other →
 
Profile Documents Logout
Upload
Lecture IX - Functorial Property of the Fundamental Group
Lecture IX - Functorial Property of the Fundamental Group

... fundamental group of a path connected space is unique upto isomorphism. However, this isomorphism is not canonical as theorem 7.9 shows and isomorphism classes of groups do not form a category. To get around this difficulty and to obtain a well-defined functor, we introduce the category of pointed t ...
open set - PlanetMath
open set - PlanetMath

Algebra II — exercise sheet 9
Algebra II — exercise sheet 9

... A1 → X, t 7→ (t2 , t3 ) is regular, birational and a homeomorphism, but not an isomorphism of varieties. Extend this to an example of a morphism of projective varieties with the same properties. Solution: The map is given by polynomials, so it is regular. It is bijective, because for any point (x, y ...
University of Bergen General Functional Analysis Problems 5 1) Let
University of Bergen General Functional Analysis Problems 5 1) Let

All the topological spaces are Hausdorff spaces and all the maps
All the topological spaces are Hausdorff spaces and all the maps

... such that p0 = ϕ◦p. The set of all morphisms from (E, p, X) to (E 0 , p0 , X) is denoted by Hom(E, E 0 ). An isomorphism ϕ from (E, p, X) to (E 0 , p0 , X) is a morphism such that ϕ : E → E 0 is a bijection whose inverse ϕ−1 : E 0 → E determines a morphism from (E 0 , p0 , X 0 ) to (E, p, X). Defini ...
Click here
Click here

Topology, Problem Set 1 Definition 1: Let X be a topological space
Topology, Problem Set 1 Definition 1: Let X be a topological space

MIDTERM 1 : Math 1700 : Spring 2014 SOLUTIONS Problem 1. (5+5
MIDTERM 1 : Math 1700 : Spring 2014 SOLUTIONS Problem 1. (5+5

... (i) Let U be an open set in Y containing f (x0 ). Since f −1 (U ) is open, there exists N such that xn ∈ f −1 (U ) for all n ≥ N . Therefore f (xn ) ∈ U for all n ≥ N , which establishes the claim. (ii) The converse is not true. Consider the topological space X = R with the topology given by the emp ...
Handout 1
Handout 1

Midterm Exam Solutions
Midterm Exam Solutions

... To prove this, first notice that R is infinite, so any two nonempty open sets in X have nontrivial intersection. Since the Hausdorff condition requires the existence of disjoint pairs of nonempty open sets, it follows that X is not Hausdorff. To see that X is compact, consider an open cover A . Choo ...
Point set topology lecture notes
Point set topology lecture notes

Topology Ph.D. Qualifying Exam Alessandro Arsie, Gerard Thompson and Mao-Pei Tsui
Topology Ph.D. Qualifying Exam Alessandro Arsie, Gerard Thompson and Mao-Pei Tsui

TOPOLOGY WEEK 3 Definition 0.1. A topological property is a
TOPOLOGY WEEK 3 Definition 0.1. A topological property is a

R -Continuous Functions and R -Compactness in Ideal Topological
R -Continuous Functions and R -Compactness in Ideal Topological

... there is a R*- neighborhood U of x such that f(U) ⊆ V. Proof: Follows from Definition 2.1. As in general topology we can prove the following theorem. Theorem 2.6 A function f: (X,τ, I)→ (Y,σ ) is R*-continuous if and only if the graph function g: X →X × Y defined by g(x) = (x, f(x)) for each x∈ X is ...
Topology Ph.D. Qualifying Exam Gerard Thompson Mao-Pei Tsui April 2009
Topology Ph.D. Qualifying Exam Gerard Thompson Mao-Pei Tsui April 2009

Disjoint unions
Disjoint unions

... Proof. We verify the universal property of a coproduct. Let Z be a topological space along with continuous maps fα : Xα → Z for all α ` ∈ A. In particular, these continuous maps are functions, so that there is a unique function f : α Xα → Z whose restrictions are f ◦ iα = fα . In other words, f is g ...
Algebraic Geometry
Algebraic Geometry

Categories and functors, the Zariski topology, and the
Categories and functors, the Zariski topology, and the

... (a) Given any category C, there is an identity functor 1C on C: it sends the object X to the object X and the morphism f to the morphism f . This is a covariant functor. (b) There is a functor from the category of groups and group homomorphisms to the category of abelian groups and homomorphisms tha ...
Qualifying Exam in Topology January 2006
Qualifying Exam in Topology January 2006

PDF
PDF

Math 541 Lecture #1 I.1: Topological Spaces
Math 541 Lecture #1 I.1: Topological Spaces

(pdf)
(pdf)

PDF
PDF

Definition. Let X be a set and T be a family of subsets of X. We say
Definition. Let X be a set and T be a family of subsets of X. We say

... (b) if Gα ∈ T for every α ∈ I, then α∈I Gα ∈ T Tn (c) if Gi ∈ T for every i ∈ {1, . . . , n}, n ∈ N, then i=1 Gi ∈ T . The sets from T are called open and their complements are called closed. Remark. Let (M, d) be a metric space. Let T be the family of all open sets in (M, d) in the sense of the the ...
PDF
PDF

< 1 ... 57 58 59 60 61 62 63 64 65 >

Sheaf (mathematics)

In mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the open sets of a topological space. The data can be restricted to smaller open sets, and the data assigned to an open set is equivalent to all collections of compatible data assigned to collections of smaller open sets covering the original one. For example, such data can consist of the rings of continuous or smooth real-valued functions defined on each open set. Sheaves are by design quite general and abstract objects, and their correct definition is rather technical. They exist in several varieties such as sheaves of sets or sheaves of rings, depending on the type of data assigned to open sets.There are also maps (or morphisms) from one sheaf to another; sheaves (of a specific type, such as sheaves of abelian groups) with their morphisms on a fixed topological space form a category. On the other hand, to each continuous map there is associated both a direct image functor, taking sheaves and their morphisms on the domain to sheaves and morphisms on the codomain, and an inverse image functor operating in the opposite direction. These functors, and certain variants of them, are essential parts of sheaf theory.Due to their general nature and versatility, sheaves have several applications in topology and especially in algebraic and differential geometry. First, geometric structures such as that of a differentiable manifold or a scheme can be expressed in terms of a sheaf of rings on the space. In such contexts several geometric constructions such as vector bundles or divisors are naturally specified in terms of sheaves. Second, sheaves provide the framework for a very general cohomology theory, which encompasses also the ""usual"" topological cohomology theories such as singular cohomology. Especially in algebraic geometry and the theory of complex manifolds, sheaf cohomology provides a powerful link between topological and geometric properties of spaces. Sheaves also provide the basis for the theory of D-modules, which provide applications to the theory of differential equations. In addition, generalisations of sheaves to more general settings than topological spaces, such as Grothendieck topology, have provided applications to mathematical logic and number theory.
  • studyres.com © 2025
  • DMCA
  • Privacy
  • Terms
  • Report