find two metric functions (distance) $\displaystyle d1 , d2$ on the space $\displaystyle V=(0,1)$ (the interval 0,1).

$\displaystyle d1 , d2$ must support:

a. $\displaystyle V$ is complete with the metric $\displaystyle d1$ and incomplete with $\displaystyle d2$.

b. $\displaystyle d1 , d2$ induce the same topology on $\displaystyle V$ (same topological space).

I apologize for any spelling mistakes and appreciate your help