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 ...
... 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 ...
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 ...
... 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 ...
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 ...
... 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 ...
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 ...
... (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 ...
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 ...
... 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 ...
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 ...
... 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 ...
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 ...
... 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 ...
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 ...
... (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 ...
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 ...
... (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 ...
