  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.
