Are there any sets that are not complete metric spaces under all possible metrics?

Question

I don't have any particular set in mind but this seemed interesthing since completeness depends on the metric.

Answer 1

Since every set can be given the discrete metric $d(x,y)=\left\{\begin{array}{ll} 1 & \text{if $x\neq y$}\\0 &\text{if $x = y$}\end{array}\right.$, and that for this metric every Cauchy sequence is stationnary and so convergent, you can't find a such example.

Answer 2

Yes, the set of rationals $\mathbb{Q}$ in its usual topology is not a Baire space, so whatever compatible metric you put on it, it will be non-complete.