It is well-known that the completion of a metric space which is homeomorphic to a ball can be very wild; in fact, I think, every compact manifold is the closure of an open ball!


But CAT(0) spaces are very different from general manifolds. If I have a CAT(0) with the path metric, that is totally bounded, and that is homeomorphic to an open ball, will its completion be a closed ball?



Is the completion of a CAT(0) open ball a closed ball?
The answer is "no".


Let $\Sigma$ be the suspension over Poincaré homology sphere.
It admits a polyhedral $\mathrm{CAT}[1]$-metric.


Let $B$ be the unit ball in the Euclidean cone $\mathrm{Cone}\,\Sigma$.


Note that $B$ is a compact $\mathrm{CAT}[0]$-space;
its interior is homeomorphic to the ball in $\mathbb{E}^5$ but its boundary is homeomorphic to $\Sigma$, which is not a manifold.


