Is this theorem about "completion of metric space" correct?

Question

It's well-known that there is a completion of a metric space unique upto isometry.

I have tried to modify this theorem slightly and I proved this statement:

Let $(X,d_X)$ be a metric space.

Then, there is a unique complete metric space $(Y,d_Y)$ such that $X\subset Y$, $d_Y \upharpoonright (X\times X)= d_X$ and $X$ is dense in $Y$.

This is indeed a yes or no question.. Since I have reformulated the theorem, I want to make sure whether I have proven it correctly.. Is this true?

Answer 1

The uniqueness of $(Y,d_Y)$ in your modified version of the theorem is certainly not true.

Precisely the point of saying that it is unique up to an isometry is that instead of $Y$ you can take any set $Y'$ such that there exists a bijection $Y\to Y'$. (And you transfer the metric form $Y$ to $Y'$ using this bijection; i.e., you take the metric on $Y'$ such that this bijection becomes isometry.)