Let $(X,d)$ and $(Y,\rho)$ be two metric spaces with a bijection between them. If $(X,d)$ is complete then can we say $(Y,\rho)$ also complete?

Question

The question arise when I study complete space . We know if there a bijection between two set then their property will same ,is it true for completeness property of a metric space. We know $\mathbb{R^2}$ and $\mathbb{C}$ has a bijection and both are complete in usual metric . Is there any other relation between metric d and ρ ?

Answer 1

The metric space of rationals $\Bbb Q$ equipped with the discrete metric is complete (Cauchy sequences are eventually constant) but the very same set $\Bbb Q$ equipped with the usual euclidean metric its not: moreover they can both be considered as metric subspaces of $\Bbb R$ .