Completeness, Closedness and Metric Reflections of Pseudometric Spaces

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 $(Y, ρ)$, we describe the maximal class $\mathbf{CEC}(Y, ρ)$ of superspaces of $(Y, ρ)$ such that $(Y, ρ)$ is complete if and only if $Y$ is closed in every $(Z, Δ) \in \mathbf{CEC}(Y, ρ)$. 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.

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