Uniformly disconnected space
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In mathematics, a uniformly disconnected space is a metric space [math]\displaystyle{ (X,d) }[/math] for which there exists [math]\displaystyle{ \lambda \gt 0 }[/math] such that no pair of distinct points [math]\displaystyle{ x,y \in X }[/math] can be connected by a [math]\displaystyle{ \lambda }[/math]-chain. A [math]\displaystyle{ \lambda }[/math]-chain between [math]\displaystyle{ x }[/math] and [math]\displaystyle{ y }[/math] is a sequence of points [math]\displaystyle{ x= x_0, x_1, \ldots, x_n = y }[/math] in [math]\displaystyle{ X }[/math] such that [math]\displaystyle{ d(x_i,x_{i+1}) \leq \lambda d(x,y), \forall i \in \{0,\ldots,n\} }[/math].[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" (in en). Analysis and Geometry in Metric Spaces 5 (1): 69–77. doi:10.1515/agms-2017-0004. ISSN 2299-3274. https://www.degruyter.com/document/doi/10.1515/agms-2017-0004/html.
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