Appendix: Basic notions and results in general topology A.1
... spaces and Q let α∈A Xα be theor cartesian product. Let (Y, U) be a topological space and f : Y → α∈A Xα a mapping. The mapping f is continuous on Y if and only if for each α ∈ A the mapping y 7→ f (y)(α) is a continuous mapping of Y to Xα . Definition. Let (X, T ) be a topological space, Y a set an ...
... spaces and Q let α∈A Xα be theor cartesian product. Let (Y, U) be a topological space and f : Y → α∈A Xα a mapping. The mapping f is continuous on Y if and only if for each α ∈ A the mapping y 7→ f (y)(α) is a continuous mapping of Y to Xα . Definition. Let (X, T ) be a topological space, Y a set an ...
THE REGULAR OPEN-OPEN TOPOLOGY FOR FUNCTION
... e X}. Then, B(Q) {W(U) V e Q} is a basis for H(X) xH(X) Q’, the quasi-uniformity of quasi-uniform convergence w.r.t. Q (Naimpally [8]). Let TO. denote the topology on H(X) induced by Q*. T0. is called the topology of quasi-uniform convergence w.r.t. Qo. If P is the Pervin quasi-uniformity on X, Tp. ...
... e X}. Then, B(Q) {W(U) V e Q} is a basis for H(X) xH(X) Q’, the quasi-uniformity of quasi-uniform convergence w.r.t. Q (Naimpally [8]). Let TO. denote the topology on H(X) induced by Q*. T0. is called the topology of quasi-uniform convergence w.r.t. Qo. If P is the Pervin quasi-uniformity on X, Tp. ...
Properties of morphisms
... Definition 2.1. Let f : X → Y be a morphism of schemes. f is called locally of finite type if Y can be covered by a collections of affine opens {Ui }i∈I such that every f −1 (Ui ) can be covered by affine opens {Vij }j∈Ji such that every f ] : OY (Ui ) → OX (Vij ) is of finite type. If all Ji can be chosen ...
... Definition 2.1. Let f : X → Y be a morphism of schemes. f is called locally of finite type if Y can be covered by a collections of affine opens {Ui }i∈I such that every f −1 (Ui ) can be covered by affine opens {Vij }j∈Ji such that every f ] : OY (Ui ) → OX (Vij ) is of finite type. If all Ji can be chosen ...
Higher Simple Homotopy Theory (Lecture 7)
... Proposition 4. Let q : E → B be a map of finite polyhedra. The following conditions are equivalent: (1) The map q is a fibration. (2) For every triangulation of B and every simplex σ of the triangulation, the induced map E ×B σ → σ is a fibration. (3) There exists a triangulation of B such that, fo ...
... Proposition 4. Let q : E → B be a map of finite polyhedra. The following conditions are equivalent: (1) The map q is a fibration. (2) For every triangulation of B and every simplex σ of the triangulation, the induced map E ×B σ → σ is a fibration. (3) There exists a triangulation of B such that, fo ...
Pointed spaces - home.uni
... the set of pointed homotopy classes of pointed maps from (X, x0 ) to (Y, y0 ). Example 3.10. Let P ' ∗ be a contractible space. For (unbased) homotopy, we have [P, Y ] ∼ = π0 (Y ) whereas for based homotopy, we have [(P, p0 ), (Y, y0 )]∗ = {∗} because every pointed map (P, p0 ) → (Y, y0 ) is pointed ...
... the set of pointed homotopy classes of pointed maps from (X, x0 ) to (Y, y0 ). Example 3.10. Let P ' ∗ be a contractible space. For (unbased) homotopy, we have [P, Y ] ∼ = π0 (Y ) whereas for based homotopy, we have [(P, p0 ), (Y, y0 )]∗ = {∗} because every pointed map (P, p0 ) → (Y, y0 ) is pointed ...
S1-Equivariant K-Theory of CP1
... dimensional vector spaces, we call E = tx∈X Ex a G -vector bundle if it is equipped with certain topological structure and a continuous action of G such that g .Ex = Eg .x . Example. (The Hopf Bundle) A canonical example of a S 1 -bundle over CP1 is H ∗ := {([d], z) ∈ CP1 × C2 : z is a point in the ...
... dimensional vector spaces, we call E = tx∈X Ex a G -vector bundle if it is equipped with certain topological structure and a continuous action of G such that g .Ex = Eg .x . Example. (The Hopf Bundle) A canonical example of a S 1 -bundle over CP1 is H ∗ := {([d], z) ∈ CP1 × C2 : z is a point in the ...
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.