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I am looking for an example of a set (X) equipped with two metrics $d$ and $\tilde{d} = \frac{d(x, y)}{1 + d(x, y)},$ where:

  1. $(X, d)$ is complete.
  2. $(X, \tilde{d})$ is not complete.

Alternatively, it could be the reverse: $(X, d)$ is not complete, but $(X, \tilde{d})$ is complete. I tried exploring this on various sets, such as:

  • $X = \mathbb{R}$,
  • $X = [0, 1]$,
  • $X = (0, 1)$,

but I have been unable to find a satisfactory example.

If anyone can provide a concrete example (with explanation), it would help clarify this concept for me.

Thank you in advance for your help!

neelkanth
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    Have you been able to determine how their sets of Cauchy sequences are related to each other? That's a pretty key starting point to understand how the completeness of the two spaces will be related. – Cameron Buie Jan 01 '25 at 17:51
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    I think these discussions answer your questions: https://math.stackexchange.com/questions/2863127/show-that-for-every-metric-d-the-metrics-d-1d-and-min-1-d-are-equ. and https://math.stackexchange.com/questions/4440234/show-that-x-d-is-complete-if-and-only-if-x-overlined-is-complete-par?rq=1 and https://math.stackexchange.com/questions/2440774/kreyszig-1-6-7-if-x-d-is-complete-show-that-x-tilded-where-tilde?rq=1. Maybe have a look. – Tri Jan 01 '25 at 17:57
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    @Tri these are equivalent but not strongly equivalent – neelkanth Jan 01 '25 at 18:02
  • @neelkanth sorry my bad. But I found this one answer your question directly: https://math.stackexchange.com/questions/1049067/x-d-is-a-complete-metric-space-iff-x-d-is-a-complete-metric-space – Tri Jan 01 '25 at 18:15
  • Whoops, yes, mis-remembered the metric. Thinking of another one. @neelkanth – Thomas Andrews Jan 01 '25 at 18:24

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