In mathematics, the Assouad–Nagata dimension (sometimes simply Nagata dimension) is a notion of dimension for metric spaces,[1][2] introduced by Jun-iti Nagata in 1958[3] and reformulated by Patrice Assouad in 1982, who introduced the now-usual definition.
[4] The Assouad–Nagata dimension of a metric space (X, d) is defined as the smallest integer n for which 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 infimum of integers k ≥ 0 such that each subset of X with diameter at most r has a non-empty intersection with at most k 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 (X, d) is the smallest integer n for which there exists a constant c > 0 such that for every r > 0, the covering of X by r-balls has a refinement with cr-multiplicity at most n + 1.