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  • $\begingroup$ There is a typo in the question that makes it difficult to understand what is being asked, but if one replaces the $d$'s with $p$'s in the bottom part of the question, then an interesting question emerges. (@youssef: I think, but I'll have to check details, that the answer is no in general, but yes if $p$ is continuous.) I'm voting to reopen. $\endgroup$
    – Will Brian
    Commented Dec 6, 2017 at 17:50
  • 1
    $\begingroup$ @WillBrian As a follow-up to your edit I have also corrected a few minor typos and explicitly added to the post that this is different from metric space. (Since this can be missed if somebody does not read carefully.) A quick Google search leads to the paper A. Amini-Harand: Metric-like spaces, partial metric spaces and fixed points, doi.org/10.1186/1687-1812-2012-204. $\endgroup$
    – Martin Sleziak
    Commented Dec 6, 2017 at 18:02
  • $\begingroup$ @ Arturo Magidin, does not reearch-level questions?!! $\endgroup$
    – youssef sabar
    Commented Dec 6, 2017 at 19:54
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    $\begingroup$ It seems that Wikipedia calls this a metametric. A reference given there is Väisälä, Jussi (2005), "Gromov hyperbolic spaces", Expositiones Mathematicae, 23 (3): 187–231, doi: 10.1016/j.exmath.2005.01.010. From this paper: "A metametric space is metrizable. In fact, a metametric $d$ can be changed to a metric $d_1$ simply by setting $d(x,x)=0$ and $d_1(x,y)=d{x,y}$ for $x\ne y$. Then $d$ and $d_1$ define the same topology." $\endgroup$
    – Martin Sleziak
    Commented Dec 7, 2017 at 1:00
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    $\begingroup$ @MartinSleziak I have cast the final vote to reopen, so that you can put some of the details from your earlier comments into an answer below $\endgroup$
    – Yemon Choi
    Commented Dec 11, 2017 at 1:04