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, p), we describe the maximal class CEC(Y, p) of superspaces of (Y, p) such that (Y, p) is complete if and only if Y is closed in every (Z, Delta) is an element of CEC(Y, p). 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. (c) 2023 Elsevier B.V. All rights reserved.