# Definition of Completion

I am learning metric spaces on my own. Recently, I am studying completion of metric space.

I have found this definition.

Is the definition correct? Should not $$Y$$ be a complete metric space?

• @drhab Are you saying that $Y$ can be a completion of $X$ even if $Y$ itself is not complete? I don't think so. Jul 2 at 11:59
• It should say "A complete metric space $(Y,d_Y)$...". So, $Y$ has to be complete by definition, and a-priori, one shouldn't use the same symbol $d$ twice. Jul 2 at 12:03
• @JoséCarlosSantos I was wrong (and of course deleted). Thank you for attending me. This made me wiser. Jul 2 at 12:04

YES. $$(Y, d')$$ should be a complete metric space .
Otherwise $$Id: X\to X$$ is an onto isomerty .
Thus $$X$$ is completion of itself. But $$(X, d)$$ need not complete here. So $$(Y, d')$$ must be a complete metric space.