For a "family" ("f""i") of pseudometrics on "X", the uniform structure defined by the family is the "least upper bound" of the uniform structures defined by the individual pseudometrics "f""i". A fundamental system of entourages of this uniformity is provided by the set of "finite" intersections of entourages of the uniformities defined by the individual pseudometrics "f""i". If the family of pseudometrics is "finite", it can be seen that the same uniform structure is defined by a "single" pseudometric, namely the "upper envelope" sup "f""i" of the family.