Assouad–Nagata dimension

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.