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,ρ), we describe the maximal class 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)∈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.