Collapsing Maps and Quasi-Isometries

We introduce a generalization of the b-metric we call a (b,c)-metric. We show that if $X$ is a $(b,c)$-metric space and $\psi: X \longrightarrow Y$ is a quasi-isometry then $Y$ is $(b,c)$-metrizable. We also define a particular kind of collapsing map that can be applied to an arbitrary $(b,c)$-metric space. We define a distance function on the image of this collapsing map and with this prove that the collapsing map is a quasi-isometry.