non-homeomorphic metric spaces with homeomorphic completions

I'm having a hard time trying to understand something.

I already know that, given two homeomorphic metric spaces, their completions aren't necessarily homeomorphic.

However, what about the opposite? I mean, is it possible for non-homeomorphic metric spaces to have homeomorphic completions?

I would like at least one example of two non-homeomorphic spaces with homeomorphic completions... I want to try to understand this better.

Thanks in advance!

Solution 1:

Yes. Take any complete metric space $M$ with some dense subset $D$ not homeomorphic to $M$. Then $M$ is homeomorphic to the completion of $D$. As an example, take $\Bbb R$ and $\Bbb Q$.