On α-normal and β-normal spaces
... Abstract. We define two natural normality type properties, α-normality and β-normality, and compare these notions to normality. A natural weakening of Jones Lemma immediately leads to generalizations of some important results on normal spaces. We observe that every β-normal, pseudocompact space is co ...
... Abstract. We define two natural normality type properties, α-normality and β-normality, and compare these notions to normality. A natural weakening of Jones Lemma immediately leads to generalizations of some important results on normal spaces. We observe that every β-normal, pseudocompact space is co ...
Rigidity of certain solvable actions on the torus
... Remark that the point x∞ = (∞, . . . , ∞) ∈ Tn is a global fixed point of the action ρB . Let G be the subgroup of GLn R consisting of linear transformations f which have the form f (x1 , . . . , xn ) = (a1 xσ(1) , . . . , an xσ(n) ) with real numbers a1 , . . . , an ̸= 0 and a permutation σ on {1, ...
... Remark that the point x∞ = (∞, . . . , ∞) ∈ Tn is a global fixed point of the action ρB . Let G be the subgroup of GLn R consisting of linear transformations f which have the form f (x1 , . . . , xn ) = (a1 xσ(1) , . . . , an xσ(n) ) with real numbers a1 , . . . , an ̸= 0 and a permutation σ on {1, ...
April 6: Groups Acting on Categories
... The main purpose of this talk is to introduce the abstract formalism of groups acting on categories, and certain related notions: weak and strong equivariance, the adjoint localization/globalization functors and convolution. We will often find ourselves in the following general setting: suppose we h ...
... The main purpose of this talk is to introduce the abstract formalism of groups acting on categories, and certain related notions: weak and strong equivariance, the adjoint localization/globalization functors and convolution. We will often find ourselves in the following general setting: suppose we h ...
Axioms of separation - GMU Math 631 Spring 2011
... Then U = i∈N Ũi and V = i∈N Ṽi are disjoint neighborhoods of H and K. 2 Exercise 15. Give an example of a Hausdorff Lindelöf space which is not regular. Theorem 16. Every Hausdorff compact space is normal. Proof. Let (X, T ) be a Hausdorff compact space, x ∈ X and H ⊂ X be a closed set such that ...
... Then U = i∈N Ũi and V = i∈N Ṽi are disjoint neighborhoods of H and K. 2 Exercise 15. Give an example of a Hausdorff Lindelöf space which is not regular. Theorem 16. Every Hausdorff compact space is normal. Proof. Let (X, T ) be a Hausdorff compact space, x ∈ X and H ⊂ X be a closed set such that ...
Takashi Noiri and Valeriu Popa THE UNIFIED THEORY
... nearly Lindelöf (resp. strongly Lindelöf), then (Y, σ) is nearly Lindelöf (resp. strongly Lindelöf). Definition 4.3. A function f : (X, mX ) → (Y, mY ) is said to be ωm-closed (resp. M-closed [15]) if f (A) is ω-m-closed (resp. mY -closed) in (Y, mY ) for every mX -closed set A of (X, mX ). Lemm ...
... nearly Lindelöf (resp. strongly Lindelöf), then (Y, σ) is nearly Lindelöf (resp. strongly Lindelöf). Definition 4.3. A function f : (X, mX ) → (Y, mY ) is said to be ωm-closed (resp. M-closed [15]) if f (A) is ω-m-closed (resp. mY -closed) in (Y, mY ) for every mX -closed set A of (X, mX ). Lemm ...
Harmonic analysis of dihedral groups
... that in which rotations act trivially, while reflections act by −1. Similarly, for non-trivial ±1-valued ψ, there is the one-dimensional subspace in which reflections act trivially, and that in which reflections act by −1. [4] When the vector space consists of functions on a set, then eigenvectors f ...
... that in which rotations act trivially, while reflections act by −1. Similarly, for non-trivial ±1-valued ψ, there is the one-dimensional subspace in which reflections act trivially, and that in which reflections act by −1. [4] When the vector space consists of functions on a set, then eigenvectors f ...
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.