Asymmetric completions of partial metric spaces

TL;DR

This paper constructs asymmetric p-Cauchy completions for all non-empty partial metric spaces, providing a positive answer to a question about the existence of such completions where the original space is dense but not symmetrically dense.

Contribution

It introduces a method to construct asymmetric p-Cauchy completions for all non-empty partial metric spaces, expanding the understanding of their completion properties.

Findings

Constructed asymmetric p-Cauchy completions for all non-empty partial metric spaces

Provided a nonstandard construction of partial metric completions

Answered positively to the open question about asymmetric density in completions

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.