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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.

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  • $\begingroup$ 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. $\endgroup$ – Arthur Feb 13 '14 at 12:07
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A generalized version of this question has been discussed here. I think the 2nd answer on the page is related to your question.

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