Hi. An absolutely closed metric space M is such that: If N is a meric space containing M, then M is closed in N. I would like to show that an absolutely closed metric space is complete, how do I do this? I know the proof of the converse but that's no help obviously. I know intuitively that an absolutely closed space should be complete: if you "imbed" the space in its completion, it will be a closed subset, and hence complete itself. But! We only know that any space is isometric to a subset of its completion, not that a completion containing the space exists (am I making sense?). The problem seems to me that for a given metric space M, you do not know if there are any other metric spaces containing it. I tried to show that the image of an absolutely closed metric space under an isometric function is itself absolutely closed, but I couldn't. If I could show that, then the image of M would be a closed subset of M's completion, and hence complete, thus M would be complete. Any help?