Is there a metric for which the open unit interval is complete?

Question

Let, $I= (0,1)$ It is well known that $I$ is not a complete with respect to the Euclidean metric $(x,y)\mapsto |x-y|$.

However, $(I,|\cdot|)$ is separable.

Question: Can we find a metric $d: I\times I \to(0,\infty)$ for which, $(I,d)$ is separable and complete?

Answer 1

Try $J = (-\pi/2,\pi/2)$ instead. How about $d(x,y) = |\tan x - \tan y|$?

Answer 2

Take a homeomorphism from the interval to $\Bbb R$ and pull back the usual Euclidean metric.