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In mathematics, a uniformly disconnected space is a metric space ${\displaystyle (X,d)}$ for which there exists ${\displaystyle \lambda >0}$ such that no pair of distinct points ${\displaystyle x,y\in X}$ can be connected by a ${\displaystyle \lambda }$-chain. A ${\displaystyle \lambda }$-chain between ${\displaystyle x}$ and ${\displaystyle y}$ is a sequence of points ${\displaystyle x=x_{0},x_{1},\ldots ,x_{n}=y}$ in ${\displaystyle X}$ such that ${\displaystyle d(x_{i},x_{i+1})\leq \lambda d(x,y),\forall i\in \{0,\ldots ,n\}}$.

Properties

Uniform disconnectedness is invariant under quasi-Möbius maps.

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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