HW1
... Here the vertical arrows are defined by pre- and post-composition with the given functions φ and ψi (how exactly?). One uses the word natural here to indicate that the isomorphism in (d) does not reflect specific properties of the spaces X, Y , and Z, but rather describes a general feature of the pr ...
... Here the vertical arrows are defined by pre- and post-composition with the given functions φ and ψi (how exactly?). One uses the word natural here to indicate that the isomorphism in (d) does not reflect specific properties of the spaces X, Y , and Z, but rather describes a general feature of the pr ...
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 ...
... 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 ...
Quiz-2 Solutions
... f : X → Y be a function. Let U be a open set in Y . Then f −1 (U ) is subset of X. So it can be written as a union of singletons, which are open subsets in the discrete metric space. Thus f −1 (U ) is a open set being a union of open subsets. Thus f is a continuous function. Consider R with the lowe ...
... f : X → Y be a function. Let U be a open set in Y . Then f −1 (U ) is subset of X. So it can be written as a union of singletons, which are open subsets in the discrete metric space. Thus f −1 (U ) is a open set being a union of open subsets. Thus f is a continuous function. Consider R with the lowe ...
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... I also write topology assignments as well. This assignment will be due on October 16th. HAHAHAHAHAAAAAA! 1. Starting from page 91, please do problems 1, 3, 4 and 6. Even though I’m a terrible boss who likes to see you all work your fingers to the bone proving topology questions, I have information a ...
... I also write topology assignments as well. This assignment will be due on October 16th. HAHAHAHAHAAAAAA! 1. Starting from page 91, please do problems 1, 3, 4 and 6. Even though I’m a terrible boss who likes to see you all work your fingers to the bone proving topology questions, I have information a ...
weak-* topology
... Let X be a locally convex topological vector space (over C or R), and let X ∗ be the set of continuous linear functionals on X (the continuous dual of X). If f ∈ X ∗ then let pf denote the seminorm pf (x) = |f (x)|, and let px (f ) denote the seminorm px (f ) = |f (x)|. Obviously any normed space is ...
... Let X be a locally convex topological vector space (over C or R), and let X ∗ be the set of continuous linear functionals on X (the continuous dual of X). If f ∈ X ∗ then let pf denote the seminorm pf (x) = |f (x)|, and let px (f ) denote the seminorm px (f ) = |f (x)|. Obviously any normed space is ...
Quotients - Dartmouth Math Home
... Math 112 : Introduction to Riemannian Geometry Quotients and Manifolds Note: The following is a brief outline of some concepts we will need in class. For more details on basic topology and quotient spaces you should see Munkres’ “Topology: A First Course”. For information on properly discontinuous a ...
... Math 112 : Introduction to Riemannian Geometry Quotients and Manifolds Note: The following is a brief outline of some concepts we will need in class. For more details on basic topology and quotient spaces you should see Munkres’ “Topology: A First Course”. For information on properly discontinuous a ...
Topology, MM8002/SF2721, Spring 2017. Exercise set 3 Exercise 1
... • Show that if X and Y are Hausdorff, so is X × Y . • Show that if X and Y are first countable, so is X × Y . • Show that if X and Y are second countable, so is X × Y . Which of these statements hold for all finite products? Which for arbitrary products? Exercise 7. Consider an arbitrary product of ...
... • Show that if X and Y are Hausdorff, so is X × Y . • Show that if X and Y are first countable, so is X × Y . • Show that if X and Y are second countable, so is X × Y . Which of these statements hold for all finite products? Which for arbitrary products? Exercise 7. Consider an arbitrary product of ...
MIDTERM EXAM
... 1. Let T and T 0 be two topologies on a set X. (a) Is their union, T ∪ T 0 , a topology on X? Why, or why not? (b) Is their intersection, T ∩ T 0 , a topology on X? Why, or why not? 2. Let p : X → Y be a continuous map. Suppose there is a continuous map f : Y → X such that p ◦ f equals the identity ...
... 1. Let T and T 0 be two topologies on a set X. (a) Is their union, T ∪ T 0 , a topology on X? Why, or why not? (b) Is their intersection, T ∩ T 0 , a topology on X? Why, or why not? 2. Let p : X → Y be a continuous map. Suppose there is a continuous map f : Y → X such that p ◦ f equals the identity ...
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... Definition 1. Let T be a category, and let C be another category. Then a i presheaf on T is a contravariant functor F : T → C. If U → V is a morphism in T , we call F (i) : F (V ) → F (U ) the restriction map obtained from i. A morphism of presheaves is a natural transformation. To understand this d ...
... Definition 1. Let T be a category, and let C be another category. Then a i presheaf on T is a contravariant functor F : T → C. If U → V is a morphism in T , we call F (i) : F (V ) → F (U ) the restriction map obtained from i. A morphism of presheaves is a natural transformation. To understand this d ...
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... Let us recall the definition of a topological group; this is a group (G, ., e) together with a topology on G such that (x, y) 7→ xy −1 is continuous, i.e., from G × G into G. Note also that G × G is regarded as a topological space defined by the product topology. Definition 0.1. Consider G to be a t ...
... Let us recall the definition of a topological group; this is a group (G, ., e) together with a topology on G such that (x, y) 7→ xy −1 is continuous, i.e., from G × G into G. Note also that G × G is regarded as a topological space defined by the product topology. Definition 0.1. Consider G to be a t ...
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 : ...
... 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 : ...
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... Then E is called a covering space, p is called a covering map, the Ei ’s are sheets of the covering of U and for each x ∈ X, p−1 (x) is the fiber of p above x. The open set U is said to be evenly covered. If E is simply connected, it is called the universal covering space. From this we can derive th ...
... Then E is called a covering space, p is called a covering map, the Ei ’s are sheets of the covering of U and for each x ∈ X, p−1 (x) is the fiber of p above x. The open set U is said to be evenly covered. If E is simply connected, it is called the universal covering space. From this we can derive th ...
