There exists a theorem (I think that it is Hausdorff theorem?) about extending subspace metrics, according to which:
Given a metric space $\ (X\ d),\ $ a closed subset $\ A\subseteq X,\ $ and a metrics $\ \rho_0\ $ in $\ A\ $ such that metrics $\ \rho_0\ $ and $\ d\,|\,A\times A\ $ are topologically equivalent (in $\ A$), there exists a metrics $\ \rho\ $ in $\ X\ $ which is topologically equivalent to $\ d,\ $ and such that $\ \rho\,|\,A\times A = \rho_0$.
An especially elegant proof was obtained by Henryk Toruńczyk.