Local metrically generated theories

In this paper we prove that for every metrically generated theory (Colebunders and Lowen, 2005 [4]) there exists a largest, what we call local metrically generated theory underlying it. We illustrate this result by looking at the traditional case of the categories of topological spaces and quasi-uniform spaces (Künzi, 2001 [10]) and also by looking at the categories of approach spaces (Lowen, 1997 [11]) and quasi-uniform gauge spaces (Lowen, 2004 [12]) and some of their usual sub- and supercategories (Choquet, 1948 [1]; Colebunders and Lowen, 1989 [2]; Herrlich, 1987 [9]; Preuss, 1988 [14]).