Assouad–Nagata dimension
In mathematics, the Assouad–Nagata dimension (sometimes simply Nagata dimension) is a notion of dimension for metric spaces, introduced by Jun-iti Nagata in 1958 and reformulated by Patrice Assouad in 1982, who introduced the now-usual definition. Definition

The Assouad–Nagata dimension of a metric space  is defined as the smallest integer  for which there exists a constant  such that for all  the space  has a -bounded covering with -multiplicity at most . Here -bounded means that the diameter of each set of the covering is bounded by , and -multiplicity is the infimum of integers  such that each subset of  with diameter at most  has a non-empty intersection with at most  members of the covering. This definition can be rephrased to make it more similar to that of the Lebesgue covering dimension. The Assouad–Nagata dimension of a metric space  is the smallest integer  for which there exists a constant  such that for every , the covering of  by -balls has a refinement with -multiplicity at most . Relationship to other notions of dimension

Compare the similar definitions of Lebesgue covering dimension and asymptotic dimension. A space has Lebesgue covering dimension at most  if it is at most -dimensional at microscopic scales, and asymptotic dimension at most  if it looks at most -dimensional upon zooming out as far as you need. To have Assouad–Nagata dimension at most , a space has to look at most -dimensional at every possible scale, in a uniform way across scales. The Nagata dimension of a metric space is always less than or equal to its Assouad dimension. References

Metric geometry
Dimension theory