A counterexample for the symmetrization of quasi-partial b-metrics

A common approach in the theory of generalized metric spaces, particularly in spaces which lack symmetry, is to pass to a symmetrized function. In the literature on quasi-partial b-metric spaces, it has been repeatedly assumed without proof that the standard symmetrization dq(x,y)=q(x,y)+q(y,x)-q(x,x)-q(y,y) is always a b-metric. We first provide a definitive counterexample to disprove this general claim. Next, from an elementary estimate derived from the axiom (QPb4), we obtain a convenient sufficient criterion: if a self-distance modulus M(q) is finite, then dq is a b-metric with explicit constant K*(q)=s+(s-1)M(q). We then show by example that this condition is not necessary. Finally, once dq is known to be a b-metric, we explain how fixed point results from complete b-metric spaces may be transferred back to the quasi-partial b-metric setting.