I'm aware that completeness and (total) boundedness are not preserved under homeomorphism, with $(0,1) \cong \mathbb R$ being a counterexample to both simultaneously. I'm curious if there exists a "double counterexample" in another way:
Are there homeomorphic metric spaces $M$ and $N$ such that
$M$ is both complete and bounded,
but $N$ is neither complete nor bounded?
Of course in that case, $M$ could not be totally bounded, else it would be compact, which is certainly a topological property!
Best Answer
You could let $M$ be $\mathbb R$ with the metric $d(x,y)=\min(1,|x-y|)$ and $N$ be $(0,\infty)$ with the usual metric.