Topological completions of metrizable spaces

Abstract:

For a pair of metrizable spaces $X$ and $Y$, we investigate conditions under which there is a dense embedding $h:X \to Z$, where $Z$ is completely metrizable and $Z\backslash h(X)$ is homeomorphic to $Y$. In such a case, $Z$ is called a topological completion of $X$ and $Y$ is called a completion remainder of $X$. In case $X$ and $Y$ are completely metrizable, we give necessary and sufficient conditions that $Y$ be a completion remainder of $X$. We characterize the completion remainders of ${\mathbf {R}}$ and those of the rationals, ${\mathbf {Q}}$. We also characterize the remainders of ${\mathbf {Q}}(\kappa )$, a nonseparable analogue of ${\mathbf {Q}}$.