I am looking for a metric on $[0,1)$ that is not complete nor totally bounded. $[0,1)$ with the Euclidean distance is not complete but totally bounded, other norms are equivalent. With the discrete metric $[0,1)$ is complete. The only idea I am left with is defining a metric with a function for example $d(x,y)=\lvert \arctan(x) - \arctan(y) \rvert$ but it not easy to guess the right one. Any ideas ?
2026-03-30 10:19:48.1774865988
Metric on $[0,1)$ that is not complete nor totally bounded
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