Let $\ (X\ d)\ \,(Y\ \delta)\ $ be arbitrary metric spaces. A function $\ f:X\rightarrow Y\ $ is called a metric map (with respect to the given metrics $\ d\ \delta$) $\ \Leftarrow:\Rightarrow\ \forall_{p\ q\in X}\ \delta(f(p)\ f(q))\ \le\ d(p\ q)$.

Injective metric spaces were introduced in a paper by Aronszajn and Panitchpakdi, under the hyper-convex spaces name, via the binary intersection property of closed balls. Equivalently, a metric space $\ (Z\ \rho)\ $ is called injective $\ \Leftarrow:\Rightarrow\ $ for every metric space $\ (X\ d)\ $ and arbitrary $\ Y\subseteq X,\ $ and for arbitrary metric map $\ f:Y\rightarrow Z\ $ (with respect to $\ \delta := d|Y\times Y$ and $\ \rho$) there exists a metric map $\ g:X\rightarrow Z\ $ (with respect to metrics $\ d\ \rho$) such that $\ g|Y=f$.

PROBLEM Characterize topologically injective metric spaces.

Preferably, this should be done for the class of all metric spaces. The class of separable spaces or of metric compact spaces would be great too.

In the case of $1$-dimensional compact spaces $\ X\ $it is pretty obvious that they are injective $\ \Leftarrow:\Rightarrow\ X$ is an AR (i.e. absolute retract as defined by Borsuk).

(Sorry, if I missed some known results (I do use Google etc, but I am terrible at searching. Please, let me know).