Completeness, Closedness and Metric Reflections of Pseudometric Spaces

TL;DR

This paper explores the conditions under which pseudometric spaces are complete and closed in their superspaces, introduces pseudoisometry, and characterizes completeness via metric reflections.

Contribution

It characterizes the maximal class of superspaces for pseudometric spaces where completeness is equivalent to closedness, and introduces pseudoisometry with a key reflection property.

Findings

Characterizes maximal classes of superspaces for completeness.

Introduces and proves properties of pseudoisometry.

Shows completeness is preserved under pseudoisometry.

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.