Uniformly disconnected space

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

Properties

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

References

^ Heinonen, Juha (2001). Lectures on Analysis on Metric Spaces. Universitext. New York: Springer-Verlag. pp. x+140. ISBN 0-387-95104-0.
^ 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.

This metric geometry-related article is a stub. You can help Wikipedia by expanding it.

[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.

[1]

[2]