A generalization of a theorem of Hurewicz for quasi-Polish spaces

We identify four countable topological spaces $S_2$, $S_1$, $S_D$, and $S_0$which serve as canonical examples of topological spaces which fail to bequasi-Polish. These four spaces respectively correspond to the $T_2$, $T_1$,$T_D$, and $T_0$-separation axioms. $S_2$ is the space of rationals, $S_1$ isthe natural numbers with the cofinite topology, $S_D$ is an infinite chainwithout a top element, and $S_0$ is the set of finite sequences of naturalnumbers with the lower topology induced by the prefix ordering. Our main resultis a generalization of Hurewicz's theorem showing that a co-analytic subset ofa quasi-Polish space is either quasi-Polish or else contains a countable$\Pi^0_2$-subset homeomorphic to one of these four spaces