On transformations of families of pseudometrics

Arbitrary transformations of a family of pseudometrics are studied which result in uniform continuous pseudometrics again. Then their properties are applied for a description of a quotient uniform structure by means of pseudometrics on the initial space. It is proved that if a uniform structure on this space is subinvariant with respect to some given set of its transformations then the quotient uniform structure is subinvariant with respect to the induced transformations. The minimization for a given family of pseudometrics is considered in the last section.