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Let $X$ be a set and let $d:X\times X\rightarrow [0,\infty)$ satisfy all the axioms of a metric besides symmetry (i.e.: $d$ is a quasi-metric). Define a topology $\tau_{d:+}$ on $X$ induced by $d$ as generated by the sub-basic open sets $\{U_{\epsilon,x}\}_{x \in X,\epsilon>0}$ where: $$ U_{\epsilon,x}:= \left\{ y \in X:\, d(x,y)<\epsilon \right\}. $$

Are there any major differences between this type of "asymmetric metric" topology and those topologies generated by regular metrics?

Details of what I mean:

  • The positive definite axiom, for me, looks like this: $d(x,y)=0 \Leftarrow x=y$; but the converse may fail. Here, I can at-least imagine that the Hausdorff property may be damaged, but I can't make things concrete...

  • Are such topologies $T_6$ (i.e.: perfectly normal Hausdorff spaces) or do they satisfy some more basic separation axioms?

Note: Is there a good reference that discusses such constructions in some depth?

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    $\begingroup$ In view of the deleted answer, should you clarify what the positive definite axiom looks like for you? Is it $d(x,y)=0$ implies $x=y$, or $d(x,y)=d(y,x)=0$ implies $x=y$? $\endgroup$
    – Nate Eldredge
    May 20, 2021 at 22:32
  • $\begingroup$ @NateEldredge (Details added) I'm only looking for a assymmetric version of the positive definite axiom. $\endgroup$
    – John_Algorithm
    May 21, 2021 at 9:28
  • $\begingroup$ It would be still better if you would write down exactly the axioms you mean. For example, I am not sure whether $d(x,y)=0$ if $x\leqslant y$ and $d(x,y)=1$ otherwise is fine for you. $\endgroup$
    – მამუკა ჯიბლაძე
    May 21, 2021 at 11:17
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    $\begingroup$ If you allow $d(x,y) = 0 \implies x=y$ to fail, then you would seem to allow the trivial "distance" given by $d(x,y)=0$ for all $x,y$. This of course induces the indiscrete topology which satisfies no separation axioms at all. If that is not what you mean, then I think you'd better write out your complete set of axioms, in full. In particular, it seems that you have in mind something rather different than what most people mean by quasi-metric $\endgroup$
    – Nate Eldredge
    May 21, 2021 at 14:00

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