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