Contents 1. Topological Space 1 2. Subspace 2 3. Continuous
... Let X and Y be topological spaces. A function f : X → Y is continuous if and only if f −1 (V ) is open in X whenever V is open in Y. A function f : X → Y is called continuous at x if for every open neighborhood of f (x), there exists an open neighborhood U of x such that f (U ) ⊂ V. Proposition 3.1. ...
... Let X and Y be topological spaces. A function f : X → Y is continuous if and only if f −1 (V ) is open in X whenever V is open in Y. A function f : X → Y is called continuous at x if for every open neighborhood of f (x), there exists an open neighborhood U of x such that f (U ) ⊂ V. Proposition 3.1. ...
CHARACTERIZATIONS OF sn-METRIZABLE SPACES Ying Ge
... is a g-metrizable space if and only if it is a quotient, π (compact), σ-image of a metric space, which improves a foregoing result on g-metrizable spaces by omitting the condition “compact-covering” in the statement. Throughout this paper, all spaces are assumed to be regular T1 , and all maps are c ...
... is a g-metrizable space if and only if it is a quotient, π (compact), σ-image of a metric space, which improves a foregoing result on g-metrizable spaces by omitting the condition “compact-covering” in the statement. Throughout this paper, all spaces are assumed to be regular T1 , and all maps are c ...
aa1
... Please study the following Problems 1-10 by January 18 (Friday). In Problems 1-10, let X be a compact Hausdorff space and let A be the ring of all C-valued continuous functions on X. We consider how X is reflected in A and how A is reflected in X like the relation of a flower and its image in the wa ...
... Please study the following Problems 1-10 by January 18 (Friday). In Problems 1-10, let X be a compact Hausdorff space and let A be the ring of all C-valued continuous functions on X. We consider how X is reflected in A and how A is reflected in X like the relation of a flower and its image in the wa ...
Proof that a compact Hausdorff space is normal (Powerpoint file)
... Since A1 is compact, (it’s a closed subset of a compact space) there exists a finite subcover {Uª1, Uª2, ... ,Uªn} such that A1(Uª1 Uª2 Uªn) Let U = Uª1, ... Uªn be the desired open set containing A1. ...
... Since A1 is compact, (it’s a closed subset of a compact space) there exists a finite subcover {Uª1, Uª2, ... ,Uªn} such that A1(Uª1 Uª2 Uªn) Let U = Uª1, ... Uªn be the desired open set containing A1. ...
1. Projective Space Let X be a topological space and R be an
... Let X be a topological space and R be an equivalence relation on X. The set of all Requivalence classes in X is denoted by X/R. The natural map π : X → X/R sending x to its equivalent class [x] is called the quotient map. We say that U ⊂ X/R is open if π −1 (U ) is open in X. This topology on X/R is ...
... Let X be a topological space and R be an equivalence relation on X. The set of all Requivalence classes in X is denoted by X/R. The natural map π : X → X/R sending x to its equivalent class [x] is called the quotient map. We say that U ⊂ X/R is open if π −1 (U ) is open in X. This topology on X/R is ...
Homework Set 3 Solutions are due Monday, November 9th.
... ii) For all x ∈ U , there is an open neighborhood W of X, with W ⊆ U , and α ∈ F(U ) such that s(x) = αx for all x ∈ W . Show that there is a map F → F + with the following universal property: for every morphism η : F → G, where G is a sheaf, there is a unique morphism of sheaves φ : F + → G such th ...
... ii) For all x ∈ U , there is an open neighborhood W of X, with W ⊆ U , and α ∈ F(U ) such that s(x) = αx for all x ∈ W . Show that there is a map F → F + with the following universal property: for every morphism η : F → G, where G is a sheaf, there is a unique morphism of sheaves φ : F + → G such th ...
Topological K-theory: problem set 7
... In particular, for a pointed space X, we can apply this to the sequence of spaces En = Σn X, where the structure maps φn are given by X → ΩΣX, the adjoint of the identity on ΣX. The resulting spectrum is called the suspension spectrum of X and is denoted by Σ∞ X. Problem 4. Let E be a spectrum. Show ...
... In particular, for a pointed space X, we can apply this to the sequence of spaces En = Σn X, where the structure maps φn are given by X → ΩΣX, the adjoint of the identity on ΣX. The resulting spectrum is called the suspension spectrum of X and is denoted by Σ∞ X. Problem 4. Let E be a spectrum. Show ...
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.