Decomposition of continuity via θ-local function in ideal topological
... Theorem 3.1. A subset A of an ideal topological space (X,τ ,I) is ∗θ-semi-I-open if and only if cl(A) = cl(int∗θ (A)). Theorem 3.2. A subset A of an ideal topological space (X,τ ,I) is ∗θ-semi-I-open if and only if for some ∗θ-open set U , U ⊆ A ⊆ cl(U ). Theorem 3.3. Let (X,τ ,I) be an ideal topolo ...
... Theorem 3.1. A subset A of an ideal topological space (X,τ ,I) is ∗θ-semi-I-open if and only if cl(A) = cl(int∗θ (A)). Theorem 3.2. A subset A of an ideal topological space (X,τ ,I) is ∗θ-semi-I-open if and only if for some ∗θ-open set U , U ⊆ A ⊆ cl(U ). Theorem 3.3. Let (X,τ ,I) be an ideal topolo ...
October 17, 2014 p-DIVISIBLE GROUPS Let`s set some conventions
... then T` (E) ' Z2` but if ` = char(K) then T` (E) is either zero or Z` . Theorem 5. Let E1 and E2 be elliptic curves over K and let ` 6= char(K) be a prime. Then Hom(E1 , E2 ) ⊗ Z` → HomGal(K/K) (T` (E1 ), T` (E2 )) is injective. Unfortunately, this fails for ` = char(K). We will try to remedy this b ...
... then T` (E) ' Z2` but if ` = char(K) then T` (E) is either zero or Z` . Theorem 5. Let E1 and E2 be elliptic curves over K and let ` 6= char(K) be a prime. Then Hom(E1 , E2 ) ⊗ Z` → HomGal(K/K) (T` (E1 ), T` (E2 )) is injective. Unfortunately, this fails for ` = char(K). We will try to remedy this b ...
bonnet theorem for open manifolds
... geo(M) < 1 for some normal covering/?: M -> Λf, M/Γ = M. The condition < 1 means that \K\ < 1 and the injectivity radius, i(X\ satisfies 1. As in [6], it actually suffices to assume geo(£/) < 1 for some neighborhood U of infinity. The details of this generalization are not difficult and will be omit ...
... geo(M) < 1 for some normal covering/?: M -> Λf, M/Γ = M. The condition < 1 means that \K\ < 1 and the injectivity radius, i(X\ satisfies 1. As in [6], it actually suffices to assume geo(£/) < 1 for some neighborhood U of infinity. The details of this generalization are not difficult and will be omit ...
MANIFOLDS, COHOMOLOGY, AND SHEAVES
... To understand these lectures, it is essential to know some point-set topology, as in [3, Appendix A], and to have a passing acquaintance with the exterior calculus of differential forms on a Euclidean space, as in [3, Sections 1–4]. To be consistent with Eduardo Cattani’s lectures at this summer sch ...
... To understand these lectures, it is essential to know some point-set topology, as in [3, Appendix A], and to have a passing acquaintance with the exterior calculus of differential forms on a Euclidean space, as in [3, Sections 1–4]. To be consistent with Eduardo Cattani’s lectures at this summer sch ...
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.