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.