Hausdorff Quasi-Polish spaces

Question

A topological space is said to be quasi-Polish if it is second-countable and completely quasi-metrizable (see for an introduction de Brecht's article de Brecht, Matthew, Quasi-Polish spaces, Ann. Pure Appl. Logic 164, No. 3, 356-381 (2013). ZBL1270.03086.) ).

These spaces have been introduced because, among other things, they allow the generalization of some results of classical descriptive set theory in non-Hausdorff environments. Now my questions are:

1. An Hausdorff quasi-Polish space needs to be Polish? Do we have a counter-example of an Hausdorff quasi-Polish space which is not Polish?
2. Is there a quasi-Polish space having a point which (whose singleton) is not closed?

Thanks!

EDIT: Here is the answer to 1.

Answer 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.