Chapter 2 Metric Spaces and Topology
... Cauchy sequences have many applications in analysis and signal processing. For example, they can be used to construct the real numbers from the rational numbers. In fact, the same approach is used to construct the completion of any metric space. Definition 2.1.33. Two Cauchy sequences x1 , x2 , . . ...
... Cauchy sequences have many applications in analysis and signal processing. For example, they can be used to construct the real numbers from the rational numbers. In fact, the same approach is used to construct the completion of any metric space. Definition 2.1.33. Two Cauchy sequences x1 , x2 , . . ...
Intro to Categories
... of a universal object. Through these definitions, we can define the product, the fiber product, and the co-product of two objects in any category. Definition 2. Let C be a category. U ∈ Obj[C] is a universally repelling object in C if for any A ∈ Obj[C] there exists a unique morphism f : U → A. It i ...
... of a universal object. Through these definitions, we can define the product, the fiber product, and the co-product of two objects in any category. Definition 2. Let C be a category. U ∈ Obj[C] is a universally repelling object in C if for any A ∈ Obj[C] there exists a unique morphism f : U → A. It i ...
RELATIONS BETWEEN UNION AND INTERSECTION OF IDEALS
... Definition 2.1. Let X be any set. An ideal in X is a nonempty collection I of subsets of X satisfying the following. i. If A, B ∈ I, then A ∪ B ∈ I. ii. If A ∈ I and B ⊆ A, then B ∈ I. If (X, T ) is a topological space and I is an ideal on X, then the triplet (X, T , I) is called an ideal topologica ...
... Definition 2.1. Let X be any set. An ideal in X is a nonempty collection I of subsets of X satisfying the following. i. If A, B ∈ I, then A ∪ B ∈ I. ii. If A ∈ I and B ⊆ A, then B ∈ I. If (X, T ) is a topological space and I is an ideal on X, then the triplet (X, T , I) is called an ideal topologica ...
Proofs - Maths TCD
... Proof. Suppose X is Hausdorff and A ⊂ X is compact. To show that X − A is open, let x ∈ X − A be given. Then for each y ∈ A there exist disjoint open sets Uy , Vy such that x ∈ Uy and y ∈ Vy . Since the sets Vy form an open cover of A, finitely many of them cover A by compactness. Suppose that Vy1 , ...
... Proof. Suppose X is Hausdorff and A ⊂ X is compact. To show that X − A is open, let x ∈ X − A be given. Then for each y ∈ A there exist disjoint open sets Uy , Vy such that x ∈ Uy and y ∈ Vy . Since the sets Vy form an open cover of A, finitely many of them cover A by compactness. Suppose that Vy1 , ...
Introduction to Topology
... continuous at a ∈ X if and only if for each ² > 0 there exists δ > 0 such that f (Bδ (a)) ⊆ B² (f (a)) (or equivalently, Bδ (a) ⊆ f −1 [B² (f (a))]). 4.3 Theorem. Let (X, d) be a metric space and let (an ) be a sequence in X. Then limn an = a if and only if for each ² > 0 there exists a positive int ...
... continuous at a ∈ X if and only if for each ² > 0 there exists δ > 0 such that f (Bδ (a)) ⊆ B² (f (a)) (or equivalently, Bδ (a) ⊆ f −1 [B² (f (a))]). 4.3 Theorem. Let (X, d) be a metric space and let (an ) be a sequence in X. Then limn an = a if and only if for each ² > 0 there exists a positive int ...
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.