Assouad–Nagata dimension

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