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 $\ g|Y=f$.

PROBLEMCharacterize topologically the toplogical spaces which are homeomorphic to the 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.*

EDITUnder a pressure from some (just one?) careful MO participants I have edited the statement of my. I must say that in my own opinion my old statement:MO-Question

.Characterize topologically injective metric spacesis much better. I would rather say

characterize topologically closed interval, circle, Euclidean plane and spherethancharacterize topologically topological spaces homeomorphic to closed interval, circle, Euclidean plane and sphere. Or one can also say simply the same using the symbols: $\ I\ S^1\ \mathbb R^2\ S^2$. The reason to me is both mathematical, as well as the simplicity of language.

category of all metric spaces (only the empty space and singletons are projective). There are no non-trivial projective objects in themetriccategory of all bounded metric spaces. And in themetriccategory of all spaces of diameter $\le 1,\ $ a space $\ (X\ d)\ $ is projective $\ \Leftarrow:\Rightarrow\ d$ is the $\ 0\!-\!1$ metric. $\endgroup$metric7more comments