Asymmetric completions of partial metric spaces

Abstract

Ge and Lin (2015) proved the existence and the uniqueness of p-Cauchy completions of partial metric spaces under symmetric denseness. They asked if every (non-empty) partial metric space $X$ has a p-Cauchy completion $\bar{X}$ such that $X$ is dense but not symmetrically dense in $\bar{X}$. We construct asymmetric p-Cauchy completions for all non-empty partial metric spaces. This gives a positive answer to the question. We also provide a nonstandard construction of partial metric completions.