Asymmetric Completions of Partial Metric Spaces
Keywords:
Cauchy completions, denseness, nonstandard analysis, partial metric spaces, symmetric densenessAbstract
Xun Ge and Shou Lin (2015) prove the existence and the uniqueness of $p$-Cauchy completions of partial metric spaces under symmetric denseness. They ask if every (non-empty) partial metric space $X$ has a $p$-Cauchy completion $\overline{X}$ such that $X$ is dense but not symmetrically dense in $\overline{X}$. 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.
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