Symmetrizations of Distance Functions and f-Quasimetric Spaces

We prove theorems on the topological equivalence of distance functions on spaces with weak and reverse weak symmetries. We study the topology induced by a distance function ρ under the condition of the existence of a lower symmetrization for ρ by an f -quasimetric. For ( q 1 , q 2 )-metric spaces ( X, ρ ), we also study the properties of their symmetrizations min { ρ ( x, y ), ρ ( y, x )} and max { ρ ( x, y ), ρ ( y, x )}. The relationship between the extreme points of a ( q 1 q 2 )-quasimetric ρ and its symmetrizations min{ ρ ( x, y ), ρ ( y, x )} and max { ρ ( x, y ), ρ ( y, x )}.