AN INTRODUCTION TO KK-THEORY These are the lecture notes of
... Proof. We only prove the last assertion here. The map T → T ⊗ 1 is linear and contractive from L(E1 , E2 ) to L(E1 ⊗ F, E2 ⊗ F ). So it suffices to consider T of the form θe2 ,e1 with e1 ∈ E1 and e2 ∈ E2 . Because E2 = E2 · B, it suffices to consider θe2 b,e1 with b ∈ B. Now for all e01 ⊗ f ∈ E1 ⊗ F ...
... Proof. We only prove the last assertion here. The map T → T ⊗ 1 is linear and contractive from L(E1 , E2 ) to L(E1 ⊗ F, E2 ⊗ F ). So it suffices to consider T of the form θe2 ,e1 with e1 ∈ E1 and e2 ∈ E2 . Because E2 = E2 · B, it suffices to consider θe2 b,e1 with b ∈ B. Now for all e01 ⊗ f ∈ E1 ⊗ F ...
On some problems in computable topology
... [31] and think of the basic open sets as easy to encode observations that can be made about the computational process determining the elements. Therefore, we let the topological basis be indexed in a total way. By doing better and better observation we want finally be able to determine every element ...
... [31] and think of the basic open sets as easy to encode observations that can be made about the computational process determining the elements. Therefore, we let the topological basis be indexed in a total way. By doing better and better observation we want finally be able to determine every element ...
as a PDF - Universität Bonn
... In the last decades, the theory of Riemann surfaces and their moduli spaces has seen a rapid development. In particular, there is now an elaborate homotopy theory of moduli spaces of Riemann surfaces. This homotopy theory of moduli spaces is based on Teichmüller theory. Teichmüller theory tells us ...
... In the last decades, the theory of Riemann surfaces and their moduli spaces has seen a rapid development. In particular, there is now an elaborate homotopy theory of moduli spaces of Riemann surfaces. This homotopy theory of moduli spaces is based on Teichmüller theory. Teichmüller theory tells us ...
CLOSED GRAPH THEOREMS FOR LOCALLY CONVEX
... direct the sum respectively of the X^ under the product and direct sum topoloaies. ...
... direct the sum respectively of the X^ under the product and direct sum topoloaies. ...
Lectures on Groups of Transformations
... a point x ∈ X is a slice such that (i)x ∈ A, (ii) G(A|A) = G(x). Note that a slice need not be a slice at any of its points. Definition . Let G be a transformation group of a space X. A normal slice is a slice A such that G(y) = G(A|A) for every y ∈ A. A regular point of X is a point at which a norm ...
... a point x ∈ X is a slice such that (i)x ∈ A, (ii) G(A|A) = G(x). Note that a slice need not be a slice at any of its points. Definition . Let G be a transformation group of a space X. A normal slice is a slice A such that G(y) = G(A|A) for every y ∈ A. A regular point of X is a point at which a norm ...
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.