Hausdorff Quasi-Polish spaces

Solution 1:

To recap the definition of the paper: a quasimetric $d: X \times X \to [0,+\infty)$ obeys the axioms

1. For all $x,y \in X: x=y \iff d(x,y)=d(y,x)=0$.
2. For all $x,y,z \in X: d(x,z) \le d(x,y) + d(y,z)$.

E.g. $X=\{0,1\}$ with $d(0,0)=0=d(1,1) = d(0,1), d(1,0)=1$ is a quasi-metric. Its topology is thus generated by the open balls $B_d(x,r), r>0$ where $B_d(x,r) = \{y \in X\mid d(x,y)<r\}$. So $B_d(0,r)= X$ for all $r>0$ while $B_d(1,1)=\{1\}$ so that $X$ gets the so-called Sierpiński topology which has a non-closed singleton $\{1\}$. This $X$ is also quasi-Polish (any sequence converges to $0$ after all), and so is an example for 2.

As to 1., I'm not yet quite sure. The Sorgenfrey line is a non-Polish quasi-metric space, but it has no countable base so cannot be quasi-Polish in that paper's definition.