Could someone specify a metric on (0,1)

[Bachelier]

could someone specify a metric on (0,1) that defines (the same topology) as the abs. value (i.e. usual) metric and makes this open interval into a complete set?

Thanks

 

[micromass]

There is a homeomorphism $T:(0,1)\rightarrow \mathbb{R}$. Pull back the metric from $\mathbb{R}$. Thus define

$$d(x,y)= |T(x)-T(y)|$$