# Metric on half-open interval s.t. subset is open w.r.t. $d$ iff open w.r.t. Euclidean metric

I wish to find a metric $d$ on the space $X = (0,1]$ such that $(X,d)$ is complete and so that a subset of $X$ is open with respect to $d$ if and only if it is open with respect to the Euclidean metric on $X$.

Any help / thoughts would be appreciated.

• Do you mean you want a subset $U\subseteq X$ to be open iff it is open as $U\subset \Bbb R$? I don't think that can be done, seing as any ball containing $1$ would be non-open in the real line. – Arthur Feb 13 '14 at 12:07