If two metric spaces are homeomorphic, do their completions have to be homeomorphic?

Question

Let $ (X_1, d_1) $ and $ (X_2, d_2) $ be metric spaces and $ (X_1^*,d_1^*), (X_2^*,d_2^*) $, respectively, their completions. If $ X_1 $ and $ X_2 $ are homeomorphic, then so are $ X_1^* $ and $ X_2^*$.

Is that statement true in general?

Answer 1

No. Take $X_1:=(0,1)$ and $X_2:=\Bbb R$, both with the standard metric.

In $X_1$ there will be Cauchy sequences converging to the endpoints, but not in $X_2$.