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Is the completion of a CAT(0) open ball a closed ball?

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! … 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? …
Brian Rushton's user avatar