I'm well aware and understand that homeomorphism do not need to preserve metric completeness, I'm just trying to work out a simple counterexample. I have tried searching around just for kicks, but only seem to find more complex ones. I'm wondering if the one I have works for it for sure?(adsbygoogle = window.adsbygoogle || []).push({});

On the interval [-pi/2, pi/2], use the sequence {an} such that an = pi/2 - 1/n. Then {an} converges to pi/2. Use the functions f(x) = arctan(x) and and f-1(x) = tan (x) as the functions for the homeomorphism. Then the fist metric would be complete since {an} is cauchy and converges in the set, but the second metric would not complete since an would not converge in the set.

Does that work? Am I off a little bit?

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# Is this a homeomorphism that does not preserve metric completeness?

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