Assouad–Nagata dimension: Difference between revisions

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In mathematics the Assouad-Nagata-dimension (or Nagata-dimension) of a metric space $X,d$ is defined as the infimum of all integers $n$ such that: There exists a constant $c>0$ such that for all $r>0$ the space $X$ has a $cr$ -bounded covering with $r$ -multiplicity at most $n+1$ . Here $cr$ -bounded means that the diameter of each set of the covering is bounded by $cr$ . And $r$ -multiplicity is the infinum of integers $n\geq 0$ such that each point belongs to at most $n$ members of the covering.

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^ Lang, Urs; Schlichenmaier, Thilo (2004-10-04). "Nagata dimension, quasisymmetric embeddings, and Lipschitz extensions". arXiv:math/0410048.

[1] Lang, Urs; Schlichenmaier, Thilo (2004-10-04). "Nagata dimension, quasisymmetric embeddings, and Lipschitz extensions". arXiv:math/0410048.

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 In mathematics the Assouad-Nagata-dimension (or Nagata-dimension) of a metric space <math>X,d</math> is defined as the
 infimum of all integers <math>n</math> such that: There exists a constant <math>c > 0 </math> such that for all <math>r > 0</math>
 the space <math>X</math> has a <math>cr</math>-bounded covering with <math>r</math>-multiplicity at most <math>n+1</math>.
+Here <math>cr</math>-bounded means that the diameter of each set of the covering is bounded by <math>cr</math>. And <math>r</math>-multiplicity is the infinum of integers <math>n \geq 0</math> such that each point belongs to at most <math>n</math> members of the covering.
-Here <math>cr</math>-bounded means that the diameter of each set of the covering is bounded by <math>cr</math>. And <math>r</math>-multiplicity
-is the infinum of integers <math>n \geq 0</math> such that each point belongs to at most <math>n</math> members of the covering.
 <ref>{{Cite journal|last=Lang|first=Urs|last2=Schlichenmaier|first2=Thilo|date=2004-10-04|title=Nagata dimension, quasisymmetric embeddings, and Lipschitz extensions|url=http://arxiv.org/abs/math/0410048|journal=arXiv:math/0410048}}</ref>