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Is there a metric for which the real numbers are not complete? Clearly this is true if there is any non-complete uncountable metric space (though I haven't been able to find an example for this neither), but I am specially interested if the metric is something that can be written with the usual functions and operations in the reals.

Fernando Chu
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  • Well, just take the usual metric on $\mathbb R$ and remove one point. – lulu Oct 30 '19 at 13:20
  • Let $f:\Bbb R\to(-1,1)$ be a homeomorpism and define $d(x,y)=|f(x)-f(y)|$. (Then $(1,2,3,\dots$) is a divergent Cauchy sequence.) – David C. Ullrich Oct 30 '19 at 13:25
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    @ThomasShelby As the OP points out (implicitly), all you need is an incomplete metric space which, as a set, is in bijection with $\mathbb R$. True, the OP refers to an "incomplete uncountable metric space", but here I think "uncountable" is intended to mean "bijective with $\mathbb R$". – lulu Oct 30 '19 at 13:28

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