λ > 0 {\displaystyle \lambda >0} 0}" loading="lazy"> such that no pair of distinct points x , y ∈ X {\displaystyle x,y\in X} can be connected by a λ {\displaystyle \lambda } -chain. A λ {\displaystyle \lambda } -chain between x {\displaystyle x} and y {\displaystyle y} is a sequence of points x = x 0 , x 1 , … , x n = y {\displaystyle x=x_{0},x_{1},\ldots ,x_{n}=y} in X {\displaystyle X} such that d ( x i , x i + 1 ) ≤ λ d ( x , y ) , ∀ i ∈ { 0 , … , n } {\displaystyle d(x_{i},x_{i+1})\leq \lambda d(x,y),\forall i\in \{0,\ldots ,n\}} .[1] "> λ > 0 {\displaystyle \lambda >0} 0}" loading="lazy"> such that no pair of distinct points x , y ∈ X {\displaystyle x,y\in X} can be connected by a λ {\displaystyle \lambda } -chain. A λ {\displaystyle \lambda } -chain between x {\displaystyle x} and y {\displaystyle y} is a sequence of points x = x 0 , x 1 , … , x n = y {\displaystyle x=x_{0},x_{1},\ldots ,x_{n}=y} in X {\displaystyle X} such that d ( x i , x i + 1 ) ≤ λ d ( x , y ) , ∀ i ∈ { 0 , … , n } {\displaystyle d(x_{i},x_{i+1})\leq \lambda d(x,y),\forall i\in \{0,\ldots ,n\}} .[1] ">

Uniformly disconnected space

In mathematics, a uniformly disconnected space is a metric space for which there exists such that no pair of distinct points can be connected by a -chain. A -chain between and is a sequence of points in such that .[1]

Properties

Uniform disconnectedness is invariant under quasi-Möbius maps.[2]

References

  1. ^ Heinonen, Juha (2001). Lectures on Analysis on Metric Spaces. Universitext. New York: Springer-Verlag. pp. x+140. ISBN 0-387-95104-0.
  2. ^ Heer, Loreno (2017-08-28). "Some Invariant Properties of Quasi-Möbius Maps". Analysis and Geometry in Metric Spaces. 5 (1): 69–77. arXiv:1603.07521. doi:10.1515/agms-2017-0004. ISSN 2299-3274.


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