In Kreyszig's Introduction to Functional Analysis we find the following theorem.
1.6-2 Theorem (Completion). For a metric space $X = (X, d)$ there exists a complete metric space $\hat X = (\hat X, \hat d)$ which has a subspace $W$ that is isometric with $X$ and is dense in $\hat X$. This space $\hat X$ is unique except for isometries.
I understand the theorem and its proof, but I wonder what the point is of the isometry between $W$ and $X$. Why can't the theorem e.g. be more simply put as
For a metric space there exists a complete metric space $\hat X=(\hat X, \hat d)$ such that $X$ is a dense subspace of $W$.
Why do we need the isometry to find a completion of $X$ that is really only a completion of an isometric space of $X$? It seems overly complicated and somewhat random to me.
It may seem overly complicated, but it's actually easier: for a metric space it usually doesn't matter what the underlying elements are. For example, if I tell you that I have a metric space with five elements, and I tell you that the metric $d$ satisfies $d(a, b) = 1 \iff a \neq b$, then you essentially already know what the metric space is. It no longer matters whether the elements are $\{1, 2, 3, 4, 5\}$ or $\{a, b, c, d, e\}$ or $\{\text{red}, \text{orange}, \text{yellow}, \text{green}, \text{blue}\}$.
In a similar vein, the completion of the rational numbers (as a metric space) are the real numbers; however, in the usual construction of the rational numbers and the real numbers, it is not technically true that $\mathbb Q \subseteq \mathbb R$. The rationals are usually equivalence classes of pairs of integers, while the real numbers -- including the real numbers that can be written as a quotient of two integers -- are some kind of infinite sets of rationals, or infinite sets of sequences of rationals. However, there is a subset of the real numbers that behaves exactly like the rationals, which is the image of the obvious inclusion $\mathbb Q \to \mathbb R$, so we usually identify the two sets.