If $W$ and $\hat{W}$ are isometric and $W$ and $\hat{W}$ dense in $V$ and $\hat{V}$ respectively, then are $V$ and $\hat{V}$ isometric?

Question

Let $V$ and $\hat{V}$ be metric spaces and $W$ and $\hat{W}$ subsets of these spaces.

If $W$ and $\hat{W}$ are isometric and $W$ and $\hat{W}$ dense in $V$ and $\hat{V}$ respectively, then are $V$ and $\hat{V}$ isometric?

Some definitions:

Two metric spaces are isometric if there is a bijection between points of the metric space and if under the bijection $T$, we have $d(x,y) = \hat{d}(Tx, Ty)$ where $T$ is the bijection.

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$A$ is dense in $B$ if for any point $x \in B$ and any neighborhood of $x$ $$B(x, \delta) = \{ y | d(x,y) < \delta \}$$ for $\delta > 0$, we have $B(x,\delta) \cap A \not = \emptyset$.

My thoughts:

We need to distinguish between two cases:

where $o_1, o_2 \in W$ and $n_1, n_2 \in X \setminus W$

where $\hat{o_1}, \hat{o_2} \in \hat{W}$ and $\hat{n_1}, \hat{n_2} \in \hat{X} \setminus \hat{W}$

1) Given $o_1$ and $n_1$, we get $\hat{o_1}$ from $T(o_1)$. We must show there is some $\hat{n_1}$ so that $$d(o_1, n_1) = d(\hat{o_1}, \hat{n_1})$$

Now map $n_1 \underbrace{\rightarrow}_{T} \hat{n_1}$

2) Given $n_1$ and $n_2$ show there are some $\hat{n_1}$ and $\hat{n_2}$ so that

$$d(n_1, n_2) = d(\hat{n_1}, \hat{n_2})$$

Now map $n_1 \underbrace{\rightarrow}_{T} \hat{n_1}$ and

$n_2 \underbrace{\rightarrow}_{T} \hat{n_2}$

The mappings must be done in such a way that $T$ remains a bijection. My trouble is to show the existence of such $\hat{n_i}$. I have not yet used this bit about being dense. Perhaps that can help.

Answer 1

The answer is negative. Take $W=\hat{W}=V=(0,1)$ and $\hat{V}=[0,1]$ (with the usual metric).

Answer 2

If $V$ is the completion of $W$ and $\hat{V}$ is the completion of $\hat{W}$, this is true. Completion is uniquely determined up to isometry by a dense subset; this is typically proven while proving that the space of equivalence classes of Cauchy sequences completes a given metric space. For reference: http://www.math.columbia.edu/~nironi/completion.pdf

I am unfamiliar with Kreyszig, but this might be the fact it relies upon?