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

I have found this definition.

enter image description here

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

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    $\begingroup$ @drhab Are you saying that $Y$ can be a completion of $X$ even if $Y$ itself is not complete? I don't think so. $\endgroup$ Jul 2 at 11:59
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    $\begingroup$ 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. $\endgroup$
    – peek-a-boo
    Jul 2 at 12:03
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    $\begingroup$ @JoséCarlosSantos I was wrong (and of course deleted). Thank you for attending me. This made me wiser. $\endgroup$
    – drhab
    Jul 2 at 12:04

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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.

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