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 $\rho$ under the condition of the existence of a lower symmetrization for $\rho$ by an $f$-quasimetric. For $(q_1,q_2)$-metric spaces $(X,\rho)$, we also study the properties of their symmetrizations $ \min\big\{\rho(x,y),\rho(y,x) \big\} $ and $\max\big\{\rho(x,y),\rho(y,x) \big\} $. The relationship between the extreme points of a $(q_1,q_2)$-quasimetric $\rho$ and its symmetrizations $ \min\!\big\{\rho(x,y),\rho(y,x)\hskip-1pt \big\} $ and $\max\big\{\rho(x,y),\rho(y,x) \big\} $.