Let $(X,d_X)$ and $(Y,d_Y)$ be metric spaces. Let $f:X\to Y$ be a uniform homeomorphism, meaning that $f$ is bijective, $f:X\to Y$ is uniformly continuous, and its inverse map $f^{-1}:Y\to X$ is uniformly continuous. Then $(X,d_X)$ is complete if and only if $(Y,d_Y)$ is complete.