The certain exact sequence of Whitehead and the classification of
... We actually need to know f ∗ already at the chain complex level, say a representative ξ∗ : C ∗ X → C ∗ Y , the existence of which is certified by the homotopy extension theorem (see [9]). We shall proceed by induction. So we first define: Definition 4.1. The map αn : X n → Y n is an n-realization of f ∗ ...
... We actually need to know f ∗ already at the chain complex level, say a representative ξ∗ : C ∗ X → C ∗ Y , the existence of which is certified by the homotopy extension theorem (see [9]). We shall proceed by induction. So we first define: Definition 4.1. The map αn : X n → Y n is an n-realization of f ∗ ...
Compact factors in finally compact products of topological spaces
... paper. For example, though the statement of Theorem 1.2 mentions final ℵn+1 -compactness only, we know just one reasonable way to prove it, that is, by splitting finally ℵn+1 -compactness into pieces of [λ, λ]-compactness, as given by Proposition 3.1: see Corollary 3.3(i). Cf. also the proofs of The ...
... paper. For example, though the statement of Theorem 1.2 mentions final ℵn+1 -compactness only, we know just one reasonable way to prove it, that is, by splitting finally ℵn+1 -compactness into pieces of [λ, λ]-compactness, as given by Proposition 3.1: see Corollary 3.3(i). Cf. also the proofs of The ...
On Some Paracompactness%type Properties of Fuzzy Topological
... Arhangel’skiii, and obtain several results about The aim of this paper is to study some them. paracompactness-type properties for fuzzy topoFirst, we give some de…nitions: logical spaces. we prove that these properties are good extensions of others de…ned by A.V.Arkhangel’skii (and studied by S.A. P ...
... Arhangel’skiii, and obtain several results about The aim of this paper is to study some them. paracompactness-type properties for fuzzy topoFirst, we give some de…nitions: logical spaces. we prove that these properties are good extensions of others de…ned by A.V.Arkhangel’skii (and studied by S.A. P ...
Introduction to Topology
... Proof. For each y ∈ Y , the set g (p −1 ({y })) is a one-point set in Z since g is constant on p −1 ({y }). Define f (y ) to be this one point. Then f : Y → Z and for each x ∈ W we have f (p(x)) = g (x). So function f exists as claimed. If f is continuous, then the composition g = f ◦ p is continuou ...
... Proof. For each y ∈ Y , the set g (p −1 ({y })) is a one-point set in Z since g is constant on p −1 ({y }). Define f (y ) to be this one point. Then f : Y → Z and for each x ∈ W we have f (p(x)) = g (x). So function f exists as claimed. If f is continuous, then the composition g = f ◦ p is continuou ...
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.