A function $d:X\times X\to \mathbb R^+$ is called quasimetric on $X$ if:

1. $d(x,y)=0$ if and only if $x=y$;
2. $d(x,z)\leq d(x,y)+d(y,z)$.

One can define a subset $A\subset (X,d)$ to be open if for each $a\in A $ there is $\epsilon>0$ such that $\{y\in X:d(a,y)<\epsilon\}$ is contained in $A$. It can be easily checked, collection of these open sets define a topology $\tau$.

Similarly, one can say a subset $A\subset (X,d)$ is open if for each $a\in A $ there is $\epsilon>0$ such that $\{y\in X:d(y,a)<\epsilon\}$ is contained in $A$. Collection of these open sets define a topology $\tau'$.

Thus, a quasimetric induces two different topologies. Is it so?