Trees and amenable equivalence relations
... In this paper we prove two less immediate results. First, we show, in the presence of a finite invariant measure, that if the equivalence relation is amenable, then a.e. equivalence class has a very simple tree-structure: THEOREM 5.1. Let (M,R) be an amenable equivalence space withfiniteR-invariant ...
... In this paper we prove two less immediate results. First, we show, in the presence of a finite invariant measure, that if the equivalence relation is amenable, then a.e. equivalence class has a very simple tree-structure: THEOREM 5.1. Let (M,R) be an amenable equivalence space withfiniteR-invariant ...
LOCALIZATION OF ALGEBRAS OVER COLOURED OPERADS
... A remarkable property of localizations in homotopy theory is the fact that they preserve many kinds of algebraic structures. That is, if a space or a spectrum X is equipped with some structure and L is a homotopical localization functor (such as, for example, localization at a set of primes, localiz ...
... A remarkable property of localizations in homotopy theory is the fact that they preserve many kinds of algebraic structures. That is, if a space or a spectrum X is equipped with some structure and L is a homotopical localization functor (such as, for example, localization at a set of primes, localiz ...
On Upper and Lower D-Continuous Multifunctions
... Y , the following statements are equivalent: (a) F is u.D.c. (l.D.c.). (b) For every open F ¾ -set V , F + (V ) (F ¡ (V )) is an open set in X. (c) For every closed G ± -set K; F ¡ (K) (F + (K)) is closed in X. (d) For each x 2 X and each net fx® g® 24 which converges to x, if V is an open F ¾ -set ...
... Y , the following statements are equivalent: (a) F is u.D.c. (l.D.c.). (b) For every open F ¾ -set V , F + (V ) (F ¡ (V )) is an open set in X. (c) For every closed G ± -set K; F ¡ (K) (F + (K)) is closed in X. (d) For each x 2 X and each net fx® g® 24 which converges to x, if V is an open F ¾ -set ...
A Poincaré inequality on loop spaces - Xue
... of the Laplacian, is related to the isoperimetric constant in Cheeger’s isoperimetric inequality µ(∂A) h = infA min{µ(A),µ(M/A)} , where the infimum is taken over all open subsets of M. Standard isoperimetric inequalities say that for an open bounded set A in Rn , the ratio between the area of its b ...
... of the Laplacian, is related to the isoperimetric constant in Cheeger’s isoperimetric inequality µ(∂A) h = infA min{µ(A),µ(M/A)} , where the infimum is taken over all open subsets of M. Standard isoperimetric inequalities say that for an open bounded set A in Rn , the ratio between the area of its b ...
REPRESENTATIONS OF DYNAMICAL SYSTEMS ON BANACH
... Fact 1.4 suggests the following general definition. Definition 1.5. Let X be a (not necessarily metrizable) compact G-space. We say that X is tame if for every element p ∈ E(X) the function p : X → X is fragmented. The class of tame dynamical systems contains the class of HNS systems and hence also ...
... Fact 1.4 suggests the following general definition. Definition 1.5. Let X be a (not necessarily metrizable) compact G-space. We say that X is tame if for every element p ∈ E(X) the function p : X → X is fragmented. The class of tame dynamical systems contains the class of HNS systems and hence also ...
Full
... Cl"-T1=2 spaces; it is not T1 . We note that every singleton f2n+1g,n 2 Z, is "IntÆCl"-open and every singleton f2mg,m 2 Z, is "IntÆCl"-closed in (Z; ). Example 5.2. Let R be an equivalence relation on Z de ned by xRy if and only if x y mod 8 and Z/8 denote the set of all equivalence classes [m], ...
... Cl"-T1=2 spaces; it is not T1 . We note that every singleton f2n+1g,n 2 Z, is "IntÆCl"-open and every singleton f2mg,m 2 Z, is "IntÆCl"-closed in (Z; ). Example 5.2. Let R be an equivalence relation on Z de ned by xRy if and only if x y mod 8 and Z/8 denote the set of all equivalence classes [m], ...
ε-Open sets
... In this section, several interesting properties and constructions of ζ -open subsets are discussed in case of anti locally countable spaces. Theorem 5 A space ( X , ℑ) is anti locally countable if and only if ( X , ℑζ ) is anti locally countable. Proof. Let A ∈ ℑζ and x ∈ A . Then by Lemma 1, there ...
... In this section, several interesting properties and constructions of ζ -open subsets are discussed in case of anti locally countable spaces. Theorem 5 A space ( X , ℑ) is anti locally countable if and only if ( X , ℑζ ) is anti locally countable. Proof. Let A ∈ ℑζ and x ∈ A . Then by Lemma 1, there ...
JK Kohli, Jeetendra Aggarwal QUASI cl
... (j) cl-continuous [18] if f −1 (V ) is open in X for every clopen set V ⊂ Y . Definitions 2.4. A topological space X is said to be (a) weakly cl-normal [23] if every pair of disjoint cl-closed subsets of X can be separated by disjoint open sets in X. (b) weakly θ-normal [11] if every pair of disjoin ...
... (j) cl-continuous [18] if f −1 (V ) is open in X for every clopen set V ⊂ Y . Definitions 2.4. A topological space X is said to be (a) weakly cl-normal [23] if every pair of disjoint cl-closed subsets of X can be separated by disjoint open sets in X. (b) weakly θ-normal [11] if every pair of disjoin ...
FURTHER DECOMPOSITIONS OF â-CONTINUITYI 1 Introduction
... Definition 2.1. A subset A of an ideal topological space (X, τ , I) is said to be an ∗ g-I-LC∗ -set if A = C ∩ D, where C is ∗ g-open and D is ∗-closed. Proposition 2.2. Let (X, τ , I) be an ideal topological space and A ⊆ X. Then the following hold: 1. If A is ∗ g-open, then A is an ∗ g-I-LC∗ -set; ...
... Definition 2.1. A subset A of an ideal topological space (X, τ , I) is said to be an ∗ g-I-LC∗ -set if A = C ∩ D, where C is ∗ g-open and D is ∗-closed. Proposition 2.2. Let (X, τ , I) be an ideal topological space and A ⊆ X. Then the following hold: 1. If A is ∗ g-open, then A is an ∗ g-I-LC∗ -set; ...
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.