Completeness preserved under homeomorphism.

Question

Let $(X,d)$ and $(Y,d')$ metric spaces and $f:X\rightarrow Y$ a homeomorphism.

if $(X,d)$ is complete, is the property of be complete preserved under $f?$ i,e. Under this conditions $Y$ must be complete too.

I think that this is not true but I can't find a counterexample.

Can someone help me please?

Answer 1

Simple example:

$\Bbb{R}$ and $(0,1)$ are homeomorphic via $$x \mapsto \frac{1}{1+2^{-x}}$$ Here $\Bbb{R}$ is complete whereas $(0,1)$ is not!