Uniformly disconnected space: Difference between revisions
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←Created page with 'A metric space <math>(X,d)</math> is called uniformly disconnected if there exists a <math>\lambda > 0</math> such that no pair of distinct points <math>x,y \in X</math> can be connected by a <math>\lambda</math>-chain. A <math>\lambda</math>-chain between <math>x</math> and <math>y</math> is a sequence of points <math>x= x_0, x_1, \ldots, x_n = y</math> in <math>X</math> such that <math>d(x_i,x_{i+1}) \leq \lambda d(x,y), \forall i \in \{0,\ldots,n\}</...' |
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such that no pair of distinct points <math>x,y \in X</math> can be connected by a <math>\lambda</math>-chain. |
such that no pair of distinct points <math>x,y \in X</math> can be connected by a <math>\lambda</math>-chain. |
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A <math>\lambda</math>-chain between <math>x</math> and <math>y</math> is a sequence of points |
A <math>\lambda</math>-chain between <math>x</math> and <math>y</math> is a sequence of points |
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<math>x= x_0, x_1, \ldots, x_n = y</math> in <math>X</math> such that <math>d(x_i,x_{i+1}) \leq \lambda d(x,y), \forall i \in \{0,\ldots,n\}</math>. |
<math>x= x_0, x_1, \ldots, x_n = y</math> in <math>X</math> such that <math>d(x_i,x_{i+1}) \leq \lambda d(x,y), \forall i \in \{0,\ldots,n\}</math>.<ref>{{cite book| last = Heinonen| first = Juha | title = Lectures on Analysis on Metric Spaces | series = Universitext | publisher = Springer-Verlag | location = New York | year = 2001 | pages = x+140 | isbn = 0-387-95104-0}}</ref> |
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==References== |
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<references/> |
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[[Category:Metric geometry]] |
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Revision as of 00:03, 19 August 2021
A metric space is called uniformly disconnected if there exists a 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]
References
- ^ Heinonen, Juha (2001). Lectures on Analysis on Metric Spaces. Universitext. New York: Springer-Verlag. pp. x+140. ISBN 0-387-95104-0.