Completeness, Closedness and Metric Reflections of Pseudometric Spaces

Abstract

It is well-known that a metric space $(X,d)$ is complete iff the set $X$ is closed in every metric superspace of $(X,d)$. For a given pseudometric space $(Y,\rho )$, we describe the maximal class $\mathbf{CEC}(Y,\rho )$ of superspaces of $(Y,\rho )$ such that $(Y,\rho )$ is complete if and only if $Y$ is closed in every $(Z,\Delta )\in \mathbf{CEC}(Y,\rho )$. We also introduce the concept of pseudoisometric spaces and prove that spaces are pseudoisometric iff their metric reflections are isometric. The last result implies that a pseudometric space is complete if and only if this space is pseudoisometric to a complete pseudometric space.

arXiv comment16 pages. arXiv admin note: text overlap with arXiv:2106.00049