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]).