Given a set "S" and a coarse structure "X", we say that the maps formula_1 and formula_2 are "close" if formula_3 is a controlled set. A subset "B" of "X" is said to be "bounded" if formula_4 is a controlled set. For coarse structures "X" and "Y", we say that formula_5 is "coarse" if for each bounded set "B" of "Y" the set formula_6 is bounded in "X" and for each controlled set "E" of "X" the set formula_7 is controlled in "Y". "X" and "Y" are said to be "coarsely equivalent" if there exists coarse maps formula_5 and formula_9 such that formula_10 is close to formula_11 and formula_12 is close to formula_13.