Extended seminorms and extended topological vector spaces
... Definition 2.2 (Topological group with operators and topological vector space). Given a group topology τ on X and a field topology θ on K, we say that 1. (X, τ ) is a topological group with operators in K if for each k ∈ K, the function ϕk is τ -τ -continuous; 2. (X, τ ) is a topological vector space ...
... Definition 2.2 (Topological group with operators and topological vector space). Given a group topology τ on X and a field topology θ on K, we say that 1. (X, τ ) is a topological group with operators in K if for each k ∈ K, the function ϕk is τ -τ -continuous; 2. (X, τ ) is a topological vector space ...
HOMOLOGICAL PROPERTIES OF NON
... / Ω; in this case, Ω is open if and only if it is open in R 2. (1, 0, 0) ∈ Ω; in that case, Ω is open if and only if Ω\{(1, 0, 0)} is open in R and φ(Ω\{(1, 0, 0)} × R) ∪ {(1, 0, 0)} is an open of S2 containing (1, 0, 0); the latter fact is possible if and only if Ω\{(1, 0, 0)} = R (otherwise, if th ...
... / Ω; in this case, Ω is open if and only if it is open in R 2. (1, 0, 0) ∈ Ω; in that case, Ω is open if and only if Ω\{(1, 0, 0)} is open in R and φ(Ω\{(1, 0, 0)} × R) ∪ {(1, 0, 0)} is an open of S2 containing (1, 0, 0); the latter fact is possible if and only if Ω\{(1, 0, 0)} = R (otherwise, if th ...
STRONGLY ZERO-PRODUCT PRESERVING MAPS ON NORMED
... B and an algebra homomorphism ϕ : A −→ B. But it is not the case in general. For some good references in the field of zero-product (Jordan zero-product) preserving maps we refer the reader to [1] and [2]. Let A and B be two normed algebras over C. We shall say that a linear map θ : A −→ B is a stron ...
... B and an algebra homomorphism ϕ : A −→ B. But it is not the case in general. For some good references in the field of zero-product (Jordan zero-product) preserving maps we refer the reader to [1] and [2]. Let A and B be two normed algebras over C. We shall say that a linear map θ : A −→ B is a stron ...
Alexandroff and Ig-Alexandroff ideal topological spaces
... Thus, (Y, σ, J) is an Ig -Alexandroff ideal space. Theorem 22. [15] Let (X, τ, I) be a T1 ideal topological space and A ⊂ X. If A is an Ig -closed set in (X, τ, I), then A is ?-closed. Theorem 23. For a T1 ideal topological space (X, τ, I), the following properties are equivalent: (1) (X, τ, I) is a ...
... Thus, (Y, σ, J) is an Ig -Alexandroff ideal space. Theorem 22. [15] Let (X, τ, I) be a T1 ideal topological space and A ⊂ X. If A is an Ig -closed set in (X, τ, I), then A is ?-closed. Theorem 23. For a T1 ideal topological space (X, τ, I), the following properties are equivalent: (1) (X, τ, I) is a ...
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.